arXiv:1701.06728v1 [math.AP] 24 Jan 2017

JARED SPECK∗† Abstract. We prove a stable shock formation result for a large class of systems of quasilinear wave equations in two spatial dimensions. We give a precise description of the dynamics all the way up to the singularity. Our main theorem applies to systems of two wave equations featuring two distinct wave speeds and various quasilinear and semilinear nonlinearities, while the solutions under study are (non-symmetric) perturbations of simple outgoing plane symmetric waves. The two waves are allowed to interact all the way up to the singularity. Our approach is robust and could be used to prove shock formation results for other related systems with many unknowns and multiple speeds, in various solution regimes, and in higher spatial dimensions. However, a fundamental aspect of our framework is that it applies only to solutions in which the “fastest wave” forms a shock while the remaining solution variables do not, even though they can be non-zero at the fastest wave’s singularity. Our approach is based on an extended version of the geometric vectorfield method developed by D. Christodoulou in his study of shock formation for scalar wave equations as well as the framework developed in our recent joint work with J. Luk, in which we proved a shock formation result for a quasilinear wave-transport system featuring a single wave operator. A key new difficulty that we encounter is that the geometric vectorfields that we use to commute the equations are, by necessity, adapted to the wave operator of the (shockforming) fast wave and therefore exhibit very poor commutation properties with the slow wave operator, much worse than their commutation properties with a transport operator. In fact, commuting the vectorfields all the way through the slow wave operator would create uncontrollable error terms. To overcome this difficulty, we rely on a first-order reformulation of the slow wave equation, which, though somewhat limiting in the precision it affords, allows us to avoid uncontrollable commutator terms. Keywords: characteristics; eikonal equation; eikonal function; genuinely nonlinear strictly hyperbolic systems; null condition; null hypersurface; singularity formation; strong null condition; vectorfield method; wave breaking Mathematics Subject Classification (2010) Primary: 35L67; Secondary: 35L05, 35L10, 35L10, 35L72

January 25, 2017 Contents 1. Introduction

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JS gratefully acknowledges support from NSF grant # DMS-1162211, from NSF CAREER grant # DMS-1454419, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology. ∗ Massachusetts Institute of Technology, Cambridge, MA, USA. [email protected]math.mit.edu. 1

Shock formation for quasilinear wave systems featuring multiple speeds 2 1.1. More precise summary of the main results 1.2. Paper outline and remarks 1.3. Further context and prior results 1.4. Additional details concerning the most relevant prior works 1.5. Remarks on the nonlinear terms and extending the results to related systems 1.6. The systems under study 1.7. Overview of the proof of the main result 2. The Remaining Ingredients in the Geometric Setup 2.1. Additional constructions related to the eikonal function 2.2. Important vectorfields, the rescaled frame, and the unit frame 2.3. Projection tensorfields, G(F rame) , and projected Lie derivatives 2.4. First and second fundamental forms, the trace of a tensorfield, covariant differential operators, and the geometric torus differential 2.5. Identities for various tensorfields 2.6. Metric decompositions 2.7. The change of variables map 2.8. Commutation vectorfields and a basic vectorfield commutation identity 2.9. Deformation tensors 2.10. Transport equations for the eikonal function quantities 2.11. Useful expressions for the null second fundamental form 2.12. Arrays of unknowns and schematic notation 2.13. Frame decomposition of the wave operator 2.14. Relationship between Cartesian and geometric partial derivative vectorfields 2.15. An algebraic expression for the transversal derivative of the slow wave 2.16. Geometric integration 2.17. Integration with respect to Cartesian forms 2.18. Comparison between the Cartesian forms and the geometric forms 3. Norms, Initial Data, Bootstrap Assumptions, and Smallness Assumptions 3.1. Norms 3.2. Strings of commutation vectorfields and vectorfield seminorms 3.3. Assumptions on the initial data and the behavior of quantities along Σ0 3.4. Initial behavior of the eikonal function quantities 3.5. T(Boot) , the positivity of µ, and the diffeomorphism property of Υ 3.6. Fundamental L∞ bootstrap assumptions 3.7. Auxiliary L∞ bootstrap assumptions 3.8. Smallness assumptions 4. Energies, Null Fluxes, and Energy-Flux Identities 4.1. Definitions of the energies and null fluxes 4.2. Coerciveness of the energy and null flux for the slow wave 4.3. Energy identities 5. Preliminary pointwise estimates 5.1. Notation for repeated differentiation 5.2. Basic assumptions, facts, and estimates that we use silently 5.3. Omission of the independent variables in some expressions 5.4. Differential operator comparison estimates

8 9 9 12 14 16 20 39 39 39 41 42 43 43 43 44 45 45 46 46 47 47 48 48 49 50 50 51 51 52 53 54 55 55 55 56 56 57 58 60 60 61 61 62

J. Speck 5.5. Pointwise estimates for Cartesian components and for the Lie derivatives of the metric on `t,u 5.6. Commutator estimates 5.7. Transport inequalities and improvements of the auxiliary bootstrap assumptions 6. L∞ Estimates Involving Higher Transversal Derivatives 6.1. Commutator estimates involving two transversal derivatives 6.2. The main estimates involving higher-order transversal derivatives 7. Sharp Estimates for µ 7.1. Sharp sup-norm and pointwise estimates for the inverse foliation density 7.2. Sharp time-integral estimates involving µ 8. Pointwise estimates 8.1. Harmless error terms 8.2. Identification of the key difficult error terms in the commuted equations 8.3. Pointwise estimates for the most difficult product 8.4. Pointwise estimates for the remaining terms in the energy estimates 9. Energy estimates and improvements of the fundamental bootstrap assumptions 9.1. Definitions of the fundamental square-integral controlling quantities 9.2. The coerciveness of the fundamental L2 -controlling quantities 9.3. The main a priori energy estimates 9.4. Estimates for the most difficult top-order energy estimate error term 9.5. L2 bounds for less degenerate top-order error integrals 9.6. Estimates involving simple error terms 9.7. Estimates for the error terms not depending on inhomogeneous terms 9.8. Proof of Prop. 9.6 10. The main stable shock formation theorem References

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Shock formation for quasilinear wave systems featuring multiple speeds 4 1. Introduction Our main goal in this paper is to develop flexible techniques that advance the theory of shock formation in initially regular solutions to quasilinear hyperbolic PDE systems featuring multiple speeds of propagation. Our techniques apply in more than one spatial dimension, a setting in which one is forced to complement the method of characteristics with an exceptionally technical ingredient: energy estimates that hold all the way up to the shock singularity. We recall that shock singularities are tied to the intersection of a family of characteristics and are such that the (initially smooth) solution remains bounded but some derivative of it blows up in finite time, a phenomenon also known as wave breaking. Our approach has robust features and could be used to prove shock formation for a large class of systems; see Subsect. 1.5 for discussion on various types of systems that could be treated with our approach. However, for convenience, we study in detail only systems of pure wave1 type in the present article. Specifically, our main result is a sharp proof of finite-time shock formation for an open set of nearly plane symmetric solutions to equations (1.6.2a)-(1.6.2b), which form a system of two quasilinear wave equations in two spatial dimensions featuring two distinct (dynamic) wave speeds; see Theorem 1.2 on pg. 8 for a rough summary of our results, Subsect. 1.6 for our assumptions on the nonlinearities, and see Theorem 10.1 for the precise statements. Here we provide a very rough summary. Theorem 1.1 (Main theorem (very rough statement)). In two spatial dimensions, under suitable assumptions on the nonlinearities2 (described in Subsect. 1.6), there exists an open3 set of regular, approximately plane symmetric4 initial data for the system (1.6.2a)-(1.6.2b) such that the solution exhibits the following behavior: the first Cartesian coordinate partial derivatives of one of the wave variables blow up in finite time while the first Cartesian coordinate partial derivatives of the other wave variable remain uniformly bounded, all the way up to the singularity in the first wave. Our proof of Theorem 10.1 is not by contradiction but is instead based on giving a complete description of the dynamics, all the way up to the first singularity, which is tied to the intersection of a family of outgoing5 approximately plane symmetric characteristics;6 see Figure 1 on pg. 22 for a picture of the setup, in which those characteristics are about to intersect to form a shock. It is important to note that our approach here relies on our assumption that the shockforming wave variable satisfies a scalar equation whose principal part depends only on the 1For

quasilinear wave equations, either the first or second Cartesian coordinate partial derivatives of the solution blow up in finite time, depending on whether the quasilinear terms are of type Φ · ∂ 2 Φ or ∂Φ · ∂ 2 Φ. 2 To ensure that shocks form, we make a genuine nonlinearity-type assumption, which results in the presence of Riccati-type terms that drive the blowup; see Remark 1.6. 3 By open, we mean open with respect to a suitable Sobolev topology. 4 By plane symmetric initial data, we mean data that are functions of the Cartesian coordinate x1 . 5 Throughout, “outgoing” roughly means right-moving, that is, along the positive x1 axis, as is shown in Figure 1. 6 The characteristics that intersect are co-dimension one and have the topology I × T, where I is an interval of time and T is the standard torus.

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shock-forming variable itself; see equation (1.6.2a).7 Especially for this reason (and for others as well), one would need new ideas to prove shock formation results in more than one spatial dimension for more general second-order quasilinear hyperbolic systems, where the coupling of all of the unknowns can occur in the principal part of all of the equations. These more complicated kinds of systems arise, for example, in the study of elasticity and crystal optics, where the unknowns are the scalar components ΦA of the solution array Φ and the evolution αβ equations can be written in the form8 HAB (∂Φ)∂α ∂β ΦB = 0 (with implied summation over α, β, and B). Prior to this paper, the only constructive proof of shock formation for a quasilinear hyperbolic system in more than one spatial dimension featuring multiple speeds was our work [16, 17], joint with J. Luk, in which we studied a system of quasilinear wave equations corresponding to a single wave operator coupled to a quasilinear transport equation.9 As we explain below, the following basic features of the system studied in [16, 17] played a crucial role in the analysis: i) there was precisely one wave operator in the system and ii) transport operators are first-order. Thus, the main contribution of the present article is that we allow for additional (second-order) quasilinear wave operators in the system. This requires new geometric and analytic ideas since, as we explain two paragraphs below, a naive approach to analyzing the additional wave operators would lead to commutator error terms that are uncontrollable near the shock. The phenomenon of shock formation is ubiquitous in the study of quasilinear hyperbolic PDEs in the sense that many systems without special structure10 are known, in the case of one spatial dimension, to admit shock-forming solutions, at least for some regular initial conditions. In fact, the theory of solutions to quasilinear hyperbolic PDE systems in one spatial dimension is rather advanced in that it incorporates the formation of shocks starting from regular initial conditions as well as their subsequent interactions, at least for solutions with small total variation. Indeed, a key reason behind the advanced status of the 1D theory is the availability of estimates within the class of functions of bounded variation; readers may consult the monograph [9] for a detailed account of the one-dimensional theory. In more than one spatial dimension, the theory of solutions to quasilinear hyperbolic PDEs (without symmetry assumptions) is much less developed, owing in part to the fact that bounded variation estimates for hyperbolic systems typically fail in this setting [23]. In fact, in more than one spatial dimension, there are very few works even on the formation of a shock11 starting from smooth initial conditions, let alone the subsequent behavior of the shock wave12 or the interaction of multiple shock waves; our work here concerns the first of 7

In contrast, the principal part of the wave equation (1.6.2b) of the non-shock-forming variable is allowed to depend on both solution variables. 8The principal coefficients H αβ (∂Φ) must, of course, verify appropriate technical assumptions to ensure AB the hyperbolicity (and local well-posedness) of the system. 9More precisely, the equations studied in [16, 17] were a formulation of the compressible Euler equations with vorticity. 10Some systems in one spatial dimension with special structure, such as totally linearly degenerate quasilinear hyperbolic systems, are not expected to admit any shock-forming solutions. 11One of course needs to make assumptions on the structure of the nonlinearities in order for shocks to form. 12We note, however, that Majda has solved [18–20], in appropriate Sobolev spaces, the shock front problem. That is, he proved a local existence result starting from an initial discontinuity given across a smooth

Shock formation for quasilinear wave systems featuring multiple speeds 6 these problems. Specifically our result builds on the body of work [1–4, 7, 10, 16, 17, 21, 22, 27] on shock formation in more than one spatial dimension, the new feature being that in the present article, we have treated wave systems with multiple wave speeds such that all solution variables are allowed to be non-zero at the first singularity. All prior shock formation results in more than one spatial dimension, with the exception of the aforementioned works [16, 17], concern scalar quasilinear wave equations, which enjoy the following fundamental property: there is only one wave speed, which is tied to the characteristics of the principal part of the equation. As we mentioned earlier, the methods of [16, 17] yield similar results for wave-transport systems in which there is precisely one wave operator. As we mentioned above, many quasilinear hyperbolic systems of mathematical and physical interest have a principal part that is more complicated than that of the scalar wave equations treated in [1–4, 7, 10, 21, 22, 27] and the wave-transport systems treated in [16, 17], which feature precisely one wave operator. It is of interest to understand whether or not shock formation also occurs in solutions to such more complicated systems in more than one spatial dimension. We now explain why our proof of shock formation for quasilinear wave systems with multiple wave speeds, though they are not the most general type of secondorder hyperbolic system of interest, requires new ideas compared to [16, 17]. Like all prior works on shock formation in more than one spatial dimension, our approach in [16, 17] was fundamentally based on the construction of geometric vectorfields adapted to the single wave operator. The following idea, originating in [1–4], lied at the heart of our analysis of [16, 17]: because the vectorfields were adapted to the wave operator, we were able to commute them all the way through it while generating only controllable error terms. Moreover, commuting all the way through the wave operator seems like an essential aspect of the proof since the special cancellations that one relies on to control error terms seem to be visible only under a covariant second-order formulation of the wave equation; see also Remark 1.1. To handle the presence of a transport operator in the system of [16, 17], we relied on the following key insight: upon introducing a geometric weight,13 one can commute the same geometric vectorfields through an essentially arbitrary,14 solution-dependent, first-order (transport) operator; thanks to the weight, one encounters only error terms that can be controlled all the way up to the shock. However, as we explain at the end of Subsect. 1.4, it seems that commuting the geometric vectorfields through a typical second-order differential operator, such as a wave operator with a speed different from the one to which the vectorfields are adapted, leads to uncontrollable error terms; see the end of Subsect. 1.4 for further discussion on this point. For this reason, our treatment of systems featuring two wave operators with strictly different speeds15 is based on the following fundamental strategy, which we discuss in more detail in Subsect. 1.4: We use a first-order reformulation of one of the wave equations (corresponding to the solution variable that does not form a shock), which, though somewhat hypersurface in the Cauchy hypersurface. The data must verify suitable jump conditions, entropy conditions, and higher-order compatibility conditions. 13Specifically, the weight is the function µ, which we describe later on in detail (see Def. 1.2). 14 More precisely, the transport operator must be transversal to the null hypersurfaces corresponding to the wave operator. 15 More precisely, the characteristics corresponding to the two wave operators are strictly separated.

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limiting in the precision it affords, allows us to commute the equations with the geometric vectorfields while avoiding uncontrollable error terms. For the solutions under consideration, the “fast wave variable,” denoted by Ψ throughout, is the one that forms a shock. That is, Ψ remains bounded but some of its first partial derivatives with respect to the Cartesian coordinates blow up. In contrast, the “slow wave variable,” denoted by w throughout, is more regular in that it has uniformly bounded first partial derivatives with respect to the Cartesian coordinates, all the way up to the shock. We can draw an analogy to the little that is known about shock formation in elasticity, a second-order hyperbolic theory in which longitudinal waves propagate at a faster speed than transverse waves [29]. In spherical symmetry, there are no transverse elastic waves, and the equations of elasticity reduce to a single scalar quasilinear wave equation that governs the propagation of longitudinal waves. In this setting, under an appropriate genuinely nonlinear assumption, John proved [12] a small-data finite-time shock formation result. Thus, for elastic waves in the simplified setting of spherical symmetry, it is precisely the fastest wave that forms a shock (while the “transverse wave part” of the solution remains trivial). In our work here, the slow moving wave w is allowed to be non-zero at the singularity. For this reason, our proof of the more regular behavior of w is non-trivial and relies on our first-order formulation of the evolution equations for w. As will become clear, one would likely need new ideas to treat data such that the slow wave variable forms the shock. This is what one expects to happen, for example, in various solution regimes for the Euler-Einstein equations of cosmology and for the Euler-Maxwell equations of plasma dynamics, where one expects the slow-moving sound waves to be able to drive finite-time shock formation in the fluid part of the system (for appropriate data). Roughly, the reason that one would need new ideas to prove shock formation for the slow wave is that in the present article, to close the energy estimates, we crucially rely on the fact that the characteristic hypersurfaces of the fast wave operator (which are also known as null hypersurfaces in view of their connection to the Lorentzian notion of a null vectorfield) are spacelike relative to the slow wave operator; indeed, this essentially defines what it means for the slow wave operator to be “slow.” We denote these fast wave characteristic hypersurfaces by Put when they are truncated at time t and we depict them in Figure 1. Analytically, the fact that the Put are spacelike relative to the slow wave operator is reflected in the coercivity estimate (4.2.1b), which shows that energy for the slow wave variable w along Put is positive definite in w and all of its partial derivatives with respect to the Cartesian coordinates and does not feature a degenerate weight. This non-degenerate coercivity along Put appears to be essential for closing the energy estimates. We remark that in one spatial dimension, one can rely exclusively on the method of characteristics (without energy estimates) and thus there are many results in which the slow wave is allowed to blow up, [8, 11, 15, 24] to name just a few. Remark 1.1 (We do not use a first-order formulation for the shock-forming wave equation). As we describe in Subsect. 1.3, it does not seem possible to treat the wave equation for Ψ using a first-order formulation in the spirit of the one that we use for w; such a formulation of the evolution equations for Ψ would not exhibit the special geometric cancellations found in our second-order formulation of it (see equation (1.6.2a) and the discussion below it).

Shock formation for quasilinear wave systems featuring multiple speeds 8 1.1. More precise summary of the main results. In our work here, we pose data on a portion of the two-dimensional spacelike Cauchy hypersurface Σ := R × T

(1.1.1)

and a portion of a null hypersurface P0 . Here and throughout, T is the standard onedimensional torus (that is, the interval [0, 1) with the endpoints identified) while “null” means null with respect to the Lorentzian metric g corresponding to the wave equation for the shock-forming variable Ψ. In fact, We tailor all geometric constructions to the fast wave metric g. See Figure 1 on pg. 22 for a picture of the setup. We allow for non-trivial data on P0 because this setup would be convenient, in principle, for proving that, at least for some of our solutions, Ψ and w are both non-zero at the singularity (roughly because one could place non-zero data on P0 near the shock). However, we do not explicitly exhibit any data such that both solution variables are guaranteed to be non-zero at the singularity. Our assumption of two spatial dimensions is for technical convenience only; similar results could be proved in three or more spatial dimensions. Our 2D assumption allows us to avoid the technical issue of deriving elliptic estimates for the foliations that we use in our analysis, which are needed in three or more spatial dimensions. The elliptic estimates would be somewhat lengthy to derive but have been well-understood in the context of shock formation starting from [4]. Our result could also be extended to different spatial topologies, though changing the spatial topology might alter the kinds of data to which our results apply.16 We recall that the shock-forming variable Ψ corresponds to the “fast speed” while the less singular variable w corresponds to the “slow speed.” We will study solutions with initial conditions that are (non-symmetric) perturbations of simple outgoing plane symmetric waves with w ≡ 0. We discuss the notion of a simple outgoing plane symmetric solution in detail in Subsubsect. 1.7.3, in which, for illustration, we provide a proof of our main results for plane symmetric solutions. By plane symmetric solutions, we mean that they depend only on the Cartesian coordinates t and x1 . In the context of our main results, the factor of T in the Cauchy hypersurface Σ = R × T corresponds to perturbations away from plane symmetry. The advantage of studying (asymmetric) perturbations of simple outgoing plane symmetric waves is that it allows us to focus our attention on the singularity formation without having to confront additional evolutionary phenomena such as dispersion, which is exhibited in the initial evolutionary phase of small-data solutions with data given on R2 . We studied similar solution regimes in our joint work [28] on shock formation for scalar quasilinear wave equations as well as our joint work [16] on the compressible Euler equations with vorticity, which we further describe below. We now give a slightly more precise statement our main results; see Theorem 10.1 for the precise statements. Theorem 1.2 (Main theorem (slightly more precise statement)). Under suitable assumptions on the nonlinearities (described in Subsect. 1.6), there exists an open set of regular initial data (belonging to an appropriate Sobolev space) for the system (1.6.2a)-(1.6.2b) such 16The

formation of the shock is local in nature. Thus, given any spatial manifold, our approach could be used to exhibit an open set of data on it such that the solution forms a shock in finite time. Roughly, there is no obstacle to proving large data shock formation on general manifolds.

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that the first Cartesian coordinate partial derivatives ∂α Ψ blow up in finite time due to the intersection the “fast” characteristics Pu , while Ψ, w, and ∂α w remain uniformly bounded, all the way up to the singularity. More precisely, we allow the data for Ψ to be large or small, but they must be close to the data corresponding to a simple plane wave. We furthermore assume that the data for w are relatively small compared to the data for Ψ. Remark 1.2. Although Theorem 1.2 refers to the Cartesian coordinates, as in all of the prior proofs of shock formation in more than one spatial dimension, we need very sharp estimates, tied to a system of geometric coordinates, to close our proof; the Cartesian coordinates are not adequate for measuring the regularity and boundedness properties of the solutions nor for tracking the intersection of characteristics. 1.2. Paper outline and remarks. • In the rest of Sect. 1, we place our main result in context, construct some of the basic geometric objects that we use in our analysis, and overview our proof. • To the extent possible, in the present paper, we cite identities and estimates proved in the work [28], which yielded similar shock formation results for scalar wave equations. • In Sect. 2, we construct the remaining geo-analytic objects that we use in our proof and exhibit their main properties. • In Sect. 3, we define the norms that we use in our analysis, state our assumptions on the initial data, and formulate bootstrap assumptions. • In Sect. 4, we define our energies and provide the energy identities that we use in our L2 analysis. • In Sects. 5-8, we derive a priori L∞ and pointwise estimates for the solution. • In Sect. 9, we derive a priori energy estimates. This is the most important and technically demanding section of the paper and it relies on all of the geometric constructions and estimates from the prior sections. • In Sect. 10, we provide our main theorem. This section is short because the estimates of the preceding sections essentially allow us to quote the proof of [28, Theorem 15.1]. 1.3. Further context and prior results. In his foundational work [25] in which he invented the Riemann invariants, Riemann initiated the rigorous study of finite-time shock formation in initially regular solutions to quasilinear hyperbolic systems in one spatial dimension, specifically the compressible Euler equations of fluid mechanics. An abundance of shock formation results in one spatial dimension were proved in the aftermath of Riemann’s work, with important contributions by Lax [15], John [11], Klainerman–Majda [13], and many others, continuing up until the present day [8]. The first constructive results on finitetime shock formation in more than one spatial dimension (without symmetry assumptions) were proved by Alinhac [1–3], who treated scalar quasilinear wave equations in two and three spatial dimensions that fail the null condition. In the case of the quasilinear wave equations of irrotational relativistic fluid mechanics (which are scalar equations), Alinhac’s results were remarkably sharpened by Christodoulou [4], whose fully geometric framework subsequently led to further shock formation results for scalar quasilinear wave equations [7, 22, 27, 28] as well as the compressible Euler equations with vorticity [16, 17]. In Subsect. 1.4, we describe some of these works in more detail.

Shock formation for quasilinear wave systems featuring multiple speeds 10 As we mentioned at the beginning, our main goal in this paper is to develop techniques for studying shock formation in solutions to quasilinear hyperbolic systems in more than one spatial dimension whose principal part exhibits multiple speeds of propagation. In any attempt to carry out such a program, one must grapple with following difficulty: the known approaches to proving shock formation in more than one spatial dimension for scalar wave equations are based on geo-analytic constructions that are fully adapted to the principal part of the scalar equation (which corresponds to a dynamic Lorentzian metric), or, more precisely, to a family of characteristic hypersurfaces corresponding to the Lorentzian metric. One might say that in prior works, all coordinate/gauge freedoms for the domain were exhausted in order to understand the intersection of the characteristics corresponding to the scalar equation and the relationship of the intersection to the blowup of the solution’s derivatives. Therefore, those works left open the question of how to prove shock formation for systems featuring multiple unknowns and a more complicated principal part with distinct speeds of propagation since, roughly speaking, one has to understand how geometric objects adapted to one part of the principal operator (that is, to one wave speed) interact with remaining part of the principal operator (corresponding to the other speeds). For systems with appropriate structure, one could skirt this difficulty by considering only a subset of initial conditions that lead to the following behavior: only one solution variable is non-zero at the first singularity. For such solutions, the analysis effectively reduces to the study of a scalar equation. A simple but important example of this approach is given by John’s study of shock formation in spherically symmetric solutions to the equations of elasticity [12], in which case the equations of motion, which generally have a complicated principal part with multiple speeds of propagation, reduce to a spherically symmetric scalar quasilinear wave equation. However, such a drastically simplified setup is mathematically and physically unsatisfying in that it is typically not stable against nontrivial perturbations with very large spatial support, no matter how small they may be. In our joint works [16, 17], we proved the first shock formation result for a system with more than one speed in which all solution variables can be active (non-zero) at the first singularity. Specifically, the system (which is actually a new formulation of the compressible Euler equations) featured one wave operator and one transport operator, and we showed that for suitable data, a family of outgoing wave characteristics intersect in finite time and cause a singularity in the first Cartesian partial derivatives of the wave variables but the not transport variable.17 With the result [16] in mind, one might say the main new contribution of this article is to upgrade the framework established in [16] in order to accommodate an additional second-order quasilinear hyperbolic scalar equation (see Subsect. 1.5 for remarks on how to extend our result to additional systems). As we mentioned earlier, our main results apply to initial data such that the “fast wave” variable Ψ (corresponding to the strictly faster of the two speeds of propagation) is the one that forms a shock. That is, Ψ remains bounded but its first Cartesian coordinate partial derivatives ∂α Ψ blow up in finite time. Like all prior works on shock formation, the blowup of ∂Ψ is driven by the presence of Riccati-type self-interaction inhomogeneous terms ∂Ψ · ∂Ψ in the wave equation for Ψ. It turns out that the “slow wave” variable w exhibits much less singular behavior, even though its wave equation is also allowed to contain, when expressed 17In

[16, 17], the transport variable was the specific vorticity, defined to be vorticity divided by density.

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relative to standard Cartesian coordinates, Riccati-type self-interaction terms ∂w · ∂w. Our ability to track the different behaviors of Ψ and w requires new geometric and analytic insights, notably the advantages that arise from our first-order reformulation of the slow wave equation. We now outline why, for the solutions under study, ∂Ψ blows up while ∂w does not, even though the wave equations for Ψ and w can have a similar structure.18 The main idea is that we consider initial data that lead to nearly simple outgoing plane wave solutions in which Ψ and w have small initial dependence on the Cartesian torus coordinate x2 and in which w and ∂w are initially relatively small and remain so throughout the evolution. By an “outgoing simple wave,” we roughly mean a solution such that w ≡ 0 and such that the dynamics of Ψ are dominated by an approximately right-moving (along the x1 axis) wave, as opposed to a combination of right- and left-moving waves. Our results show that for (non-symmetric) perturbations of such solutions,19 ∂Ψ blows up before the small terms ∂w · ∂w are able to drive the blowup of ∂w. More precisely, in the solution regime under consideration, w and ∂w remain small in L∞ all the way up to the first singularity in ∂Ψ. This is a subtle effect in that the wave equation for w is allowed to contain source terms (see RHS (1.6.2b)) that are linear (but not higher order!) in the tensorial component of ∂Ψ that blows up. Therefore, our work entails the study of non-trivial interactions between a wave that forms a singularity with another wave that, as we must prove, exhibits less singular behavior, even though it is coupled to the “singular part” of the singular wave. The set of initial data that we treat in our main results is motivated in part by John’s aforementioned work [11], in which he proved a blowup result for first-order quasilinear hyperbolic systems with multiple speeds in one spatial dimension whose small-data solutions behave like simple waves20 near the singularity. John’s result was recently sharpened [8] by Christodoulou–Perez using extensions of the framework developed by Christodoulou in [4]. In Subsubsect. 1.7.3, to illustrate some of the main ideas, we provide a proof of our main results in the special case that the initial data have exact plane symmetry, that is, for data that depend only on the Cartesian coordinates t and x1 . We caution, however, that the assumption of plane symmetry represents a drastic simplification of the full problem; in plane symmetry, we are able to avoid deriving energy estimates, which is the main technical difficulty that one encounters in the general case. As we will explain, our analytic approach, in particular our approach to energy estimates, is based on geometric decompositions adapted to the characteristics corresponding to principal part of Ψ’s wave equation together with a first-order reformulation of the wave equation for w that is compatible with these geometric decompositions. This allows us to track the distinct behavior of the two waves all the way to the shock. As we mentioned above, one would likely need new ideas to treat data such that slow wave ∂w is expected to form the first singularity and, at the same time, to interact with the fast wave Ψ and its derivatives. 18In

fact, the wave equation for w is allowed to be even more complicated than that of Ψ since we allow the principal operator for w to depend on Ψ, w, and ∂w, while the principal operator for Ψ is allowed to depend only on Ψ; see (1.6.2a)-(1.6.2b). 19Actually, the equations under study do not generally admit simple plane wave solutions, though they admit solutions that are plane symmetric and nearly simple. 20That is, in [11], one non-zero solution variable is dominant near the singularity.

Shock formation for quasilinear wave systems featuring multiple speeds 12 1.4. Additional details concerning the most relevant prior works. Our work builds upon the outstanding contributions of Alinhac [1–3], who was the first to prove small-data shock formation in quasilinear wave equations in more than one spatial dimension. Specifically, Alinhac studied scalar quasilinear wave equations of the form (g −1 )αβ (∂Φ)∂α ∂β Φ = 0 on R1+n , n = 2, 3. For all such equations that fail the null condition, he identified a set of small, regular, compactly supported initial data such that the corresponding solution forms a shock in finite time. The set of data to which his main results apply were such that the constant-time hypersurface of first blowup contains exactly one point at which ∂ 2 Φ blows up.21 The most important ingredient in Alinhac’s approach was a system of dynamic geometric coordinates tied to an eikonal function, whose level sets are g-null hypersurfaces (that is, characteristics). The main idea behind his approach was as follows: show that relative to the geometric coordinates, the solution remains regular up to the singularity, except possibly at the very high (geometric) derivative levels. This enables one to approach the problem of shock formation from a more traditional perspective in which one derives long-time-existence-type estimates. It turns out that this approach, while viable, is extremely technically demanding to implement. The reason is that the best estimates known allow for the possibility that the high-order geometric energies blow up, which makes it difficult (though possible) to prove that the solution remains regular relative to the geometric coordinates at the lower derivative levels. After one has obtained regular estimates (at the lower derivative levels) relative to the geometric coordinates, one is ready the easy part of the proof: deriving the singularity formation relative to the Cartesian coordinates, which one obtains as a consequence of a degeneration of the geometric coordinates relative to the Cartesian ones. Roughly, both the formation of the shock and the blowup of the solution’s Cartesian partial derivatives are caused by the intersection of the level sets of the eikonal function. The shock-generating initial conditions identified by Alinhac form a set of “non-degenerate” data, which may be thought of as generic. Although Alinhac’s use of an eikonal function allowed him to provide a sharp description of the first singularity, his approach to deriving energy estimates was based on a Nash–Moser iteration scheme featuring a free boundary. His iteration scheme fundamentally relied on his non-degeneracy assumptions on the data, which led to solutions whose first singularity is isolated in the constant-time hypersurface of first blowup. His reliance on a Nash–Moser scheme is tied to the fact that the regularity theory for the eikonal function is very difficult at the top-order. Our work also builds upon Christodoulou’s groundbreaking sharpening [4] of Alinhac’s results for the subclass of (scalar) Euler-Lagrange wave equations corresponding to the irrotational relativistic Euler equations in three spatial dimensions. For these wave equations, Christodoulou proved the following main results: i) He showed that for solutions generated by small,22 regular, compactly supported data, shocks are the only possible singularities. The formation of a shock exactly corresponds to the intersection of the characteristics, or equivalently, the vanishing of the inverse foliation density of the characteristics, denoted by µ (see Def. 1.2). Put differently, he proved that if the characteristics never intersect, then the 21For

equations of type (g −1 )αβ (∂Φ)∂α ∂β Φ = 0, a shock singularity is such that Φ and ∂Φ remain bounded while ∂ Φ blows up. 22In particular, unlike Alinhac’s proof, Christodoulou’s yields information about an open set of data that contains the trivial data in its interior. 2

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solution exists globally.23 ii) He exhibited an open set of data for which shocks form. Unlike Alinhac’s data, Christodoulou’s do not have to lead to a solution with an isolated first singularity. Most importantly, iii) He gave a complete description of a portion of the maximal development24 of the data, all the way up to the boundary. Although for brevity we have not given a complete description of the maximal development in this article, we likely could use our sharp estimates to do so, by invoking arguments along the lines of [4, Chapter 15]. Like Alinhac’s approach and the approach of the present article, Christodoulou’s framework was based on an eikonal function (see Subsubsect. 1.7.1 for a precise definition). However, unlike Alinhac, Christodoulou did not use Nash–Moser estimates when deriving energy estimates. This required him to sharply understand the tensorial regularity properties of eikonal functions in the context of shock formation, in which their level sets intersect. In this endeavor, he was undoubtedly aided by the experience he gained from his celebrated joint proof with Klainerman of the stability of Minkowski spacetime [5]. In that work, the authors also had to deeply understand the high-order regularity properties of eikonal functions, though in a less degenerate context in which they remain regular. Christodoulou’s sharp description of the maximal development, though of interest in itself, is also important for another reason: it is an essential ingredient for properly setting up the shock development problem, which is the problem of weakly continuing the solution to the relativistic Euler equations past the first singularity under appropriate selection criteria in the form of jump conditions; see [4] for further discussion. We note that the shock development problem was recently solved in spherical symmetry [6]. Christodoulou’s sharp, geometric approach has led to further advancements on shock formation in solutions starting from smooth initial conditions, including extensions of his results to larger classes of equations and new types of initial conditions [7,10,16,17,21,22,27]; see the survey article [10] for an in-depth discussion of some of these results. However, a crucial feature of both Alinhac’s and Christodoulou’s frameworks are that they are tailored precisely to a single family of characteristics – the family whose intersection corresponds to a shock singularity. Thus, all prior works left open the question of whether or not these approaches can be adapted to systems featuring multiple speeds of propagation. As we mentioned above, first affirmative result in this direction was provided by our joint works [16, 17] with J. Luk, in which we discovered some remarkable geo-analytic structures in the compressible Euler equations with vorticity. Inspired by these structures, we developed an extended version of Christodoulou’s framework that we used to prove shock formation for solutions to a quasilinear wave-transport system. More precisely, the wave-transport system that we studied in [16, 17] was a new formulation of the compressible Euler equations, where the velocity and density satisfied a system of covariant wave equations, all with the same covariant wave operator g (corresponding to a single Lorentzian metric g), and the vorticity satisfied a (first-order) transport equation. There were two speeds in the system: the speed of sound, corresponding to sound wave propagation and the speed associated to the transporting of vorticity. A particularly remarkable aspect of the equations studied in [16, 17], which is central to the proofs, is that the inhomogeneous terms had a good null structure that did not 23More

precisely, he showed that if the characteristics never intersect, then the solution is global in a region trapped in between an inner null cone and an outer null cone. 24Roughly, the maximal development is the largest possible classical solution that is uniquely determined by the data. Readers may consult [26, 30] for further discussion.

Shock formation for quasilinear wave systems featuring multiple speeds 14 interfere with the shock formation processes. The null structures, which are fully nonlinear in nature, are a tensorial generalization of the good null structure enjoyed by the standard null form Qg (∂Ψ, ∂Ψ), which is an admissible term in our systems (1.6.2a)-(1.6.2b) (see just below those equations for further discussion on this point). In our works [16, 17], to control the wave-transport solution’s derivatives, we followed the approach of [4] (itself inspired by the Christodoulou–Klainerman proof [5] of the stability of Minkowski spacetime) and constructed a family of dynamic geometric objects, including geometric vectorfields, adapted to the characteristics25 of g. A seemingly unavoidable aspect of our approach in [16, 17] was that, due to the coupled nature of the system, we were forced to commute the transport equation with the same geometric vectorfields in order to obtain estimates for the solution’s derivatives. In general, one might expect to encounter crippling error terms from this procedure, since the geometric vectorfields are not adapted to the transport operator. What allowed our proof to go through are the following facts: i) transport operators are first-order and ii) the operators µ∂α exhibit good commutation properties with the geometric vectorfields, where ∂α is a Cartesian coordinate partial derivative vectorfield and µ > 0, mentioned above, is the inverse foliation density of the wave characteristics (which we rigorously define in Subsubsect. 1.7.1 since µ plays a critical role in the present work as well). Therefore, since the transport operator was just a (solutiondependent) linear combination of the ∂α , upon multiplying the transport equation by µ and commuting it with the geometric vectorfields, we were able to completely avoid the worst imaginable commutator error terms, which enabled us to close the proof. We now stress that our approach in [16, 17] does not allow one to commute the geometric vectorfields through typical second-order operators µ∂α ∂β ; typically one generates crippling commutator error terms26 featuring a factor of 1/µ, which blows up as µ → 0 and obstructs the approach of deriving regular estimates. In particular, the approach of [16, 17], in itself, does not manifestly allow one to couple an additional quasilinear wave equation to the system corresponding to a metric h different from g. Here it makes sense to clarify the following point: the crippling error terms do not arise when commuting the geometric vectorfields through g (since the vectorfields are adapted to the characteristics of g), but they do arise when commuting them through a typical second-order differential operator. Thus, the works [16, 17] left open the question of how to prove shock formation for solutions to second-order quasilinear systems featuring more than one wave operator. As we have mentioned, in the present article, we prove the first shock formation results for systems of this type. The following key idea, mentioned earlier, lies at the heart of our approach here. It is possible to formulate the wave equation for the non-shock-forming slow variable as a first-order system that can be treated using an extension of the approach of [16]; see equations (1.6.11a)-(1.6.11d). 1.5. Remarks on the nonlinear terms and extending the results to related systems. The formation of shocks exhibited by Theorem 1.2 is of course tied to our structural 25More

precisely, as we describe in Subsect. 1.7, the vectorfields were adapted to an eikonal function corresponding to g, whose level sets are g-null hypersurfaces. 26Using a weight with a different power of µ, such as µ2 ∂ ∂ , also seems to lead to insurmountable α β difficulties.

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assumption on the nonlinearities, which we precisely describe in Subsect. 1.6. As we mentioned earlier, the blowup of Ψ is driven by the presence of a Riccati-type interaction term in its wave equation, as is captured by our assumption (1.6.9) below. For this reason, the wave equation of Ψ may be caricatured as27 L(F lat) ∂1 Ψ ∼ (∂1 Ψ)2 + Error, where L(F lat) := ∂t + ∂1 and Error depends on Ψ, w, and their derivatives (and in particular contains the quasilinear interaction terms). Although this caricature wave equation suggests that ∂1 Ψ should blow up in finite time along the integral curves of L(F lat) , this is not how our proof works. It seems that in order to close the energy estimates and to show that error terms do not interfere with the blowup, one needs to derive very sharp estimates tailored to the family of characteristics corresponding to Ψ, which are in turn influenced by Ψ in view of the quasilinear nature28 of the equation. Another limitation of the caricature wave equation is that it suggests that Ψ itself should blow up, which is not the case for the solutions covered by our main theorem. We also stress that as in other shock formation results, our proof is more sensitive to perturbations of the equations than typical proofs of global existence. This is not surprising in view of the fact that adding terms of the form, say ±(∂1 Ψ)3 , to the RHS of the above caricature equation can drastically change the nature of its solutions. In contrast, since w and ∂α w remain bounded up to the shock, our approach is able to accommodate essentially arbitrary semilinear terms comprised of products of these variables; see Subsect. 1.6 for precise assumptions. For convenience, we have chosen not to treat the most general type of system to which our approach applies. Our approach is flexible in the sense that it could be used to treat systems featuring additional wave equations, transport equations, or symmetric hyperbolic equations. However, the following assumptions play a critical role in our analysis. • The Lorentzian metric g corresponding to the principal part of the wave equation of Ψ depends only on Ψ. That is, in the wave equation (1.6.2a) below, g = g(Ψ). More generally, we could allow for g = g(Ψ1 , · · · , Ψm ) as long as the same metric g corresponds to the principal part of the wave equation of Ψi for 1 ≤ i ≤ m. This assumption is needed to control the top-order derivatives of the eikonal function corresponding to g (see the discussion of modified quantities in Subsubsect. 1.7.6 for more details on this point). • The shock-forming variable Ψ corresponds to the fastest speed (in the strict sense) in the system. This assumption implies that all null hypersurfaces corresponding to the metric g(Ψ) are spacelike from the perspective of the principal parts of the remaining equations in the system. This is important because our proof requires the availability of positive definite energies for the slow solution variables along g-null hypersurfaces. In the present article, the positive energies for the slow wave w are guaranteed by the estimates in equation (4.2.1b). • See Remark 1.5 below for a discussion of other types of “fast” wave equations that we could treat.

27Throughout,

if V is a vectorfield and f is a scalar function, then V f := V α ∂α f denotes the V -directional

derivative of f. 28The metric g in the wave equation for Ψ verifies g = g(Ψ). In particular, g does not depend on w and thus the characteristics corresponding to g are not directly influenced by w.

Shock formation for quasilinear wave systems featuring multiple speeds 16 Remark 1.3. In Subsect. 1.6 we will make further assumptions on the nonlinearities and quantify the assumption that Ψ is the fast wave. 1.5.1. Basic notational and index conventions. We now summarize some our notation. Some of the concepts referred to here are defined later in the article. Throughout, {xα }α=0,1,2 denote the standard Cartesian coordinates on spacetime R × Σ, where x0 ∈ R is the time variable and (x1 , x2 ) ∈ Σ = R × T are the space variables. We denote the corresponding ∂ partial derivative vectorfields by ∂α =: (which are globally defined and smooth even ∂xα 2 though x is only locally defined) and we often use the alternate notation t := x0 and ∂t := ∂0 . • Lowercase Greek spacetime indices α, β, etc. correspond to the Cartesian spacetime coordinates and vary over 0, 1, 2. Lowercase Latin spatial indices a,b, etc. correspond to the Cartesian spatial coordinates and vary over 1, 2. We use tilded indices such as α e in the same way that we use their non-tilded counterparts. All lowercase Greek indices are lowered and raised with the fast wave spacetime metric g and its inverse g −1 , and not with the Minkowski metric. • We use Einstein’s summation convention in that repeated indices are summed over their respective ranges. • We sometimes use · to denote the natural contraction between two tensors (and thus raising or lowering indices with a metric is not needed). For example, if ξ is a spacetime one-form and V is a spacetime vectorfield, then ξ · V := ξα V α . • If ξ is a one-form and V is a vectorfield, then ξV := ξα V α . Similarly, if W is a vectorfield, then WV := Wα V α = g(W, V ). • If ξ is an `t,u -tangent one-form (as defined in Subsect. 2.3), then ξ # denotes its g/-dual vectorfield, where g/ is on `t,u by g. Similarly, if ξ is the Riemannian metric induced 0 1 # a symmetric type 2 `t,u -tangent tensor, then ξ denotes the type 1 `t,u -tangent tensor formed by raising one index with g/−1 and ξ ## denotes the type 20 `t,u -tangent tensor formed by raising both indices with g/−1 . • If ξ is an `t,u -tangent tensor, then the norm |ξ| is defined relative to the Riemannian metric g/, as in Def. 3.1. • Unless otherwise indicated, all quantities in our estimates that are not explicitly under an integral are viewed as functions of the geometric coordinates (t, u, ϑ) of Def. 2.2. Unless otherwise indicated, quantities under integrals have the functional dependence established below in Def. 2.24. • If Q1 and Q2 are two operators, then [Q1 , Q2 ] = Q1 Q2 − Q2 Q1 denotes their commutator. • A . B means that there exists C > 0 such that A ≤ CB. • A = O(B) means that |A| . |B|. • Constants such as C and c are free to vary from line to line. Explicit and implicit constants are allowed to depend in an increasing, continuous fashion on the data-size parameters ˚ δ and ˚ δ−1 ∗ from Subsect. 3.3. However, the constants can be chosen to be independent of the parameters ˚ and ε whenever ˚ and ε are sufficiently small relative to ˚ δ−1 and ˚ δ∗ . • b·c and d·e respectively denote the standard floor and ceiling functions. 1.6. The systems under study.

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1.6.1. Statement of the equations. For notational convenience, we introduce the following array associated to the slow wave: ~ := (w, w0 , w1 , w2 ), W wα := ∂α w, (α = 0, 1, 2), (1.6.1) where we again stress that ∂α denotes a Cartesian coordinate partial derivative vectorfield. ~ ). Throughout the article, Remark 1.4 (Remark on the pointwise norm of the array W ~ (see (1.6.1)) to be an array of scalar functions without tensorial structure. Thus, we view W ~ |2 := w2 + P2 w2 with the definition there should be no confusion of the definition |W α=0 α (3.1.1) below for the pointwise norm of an `t,u -tangent tensor.

The system of wave equations under study Our main results concern the following system of two wave equations: ~ )Qg (∂Ψ, ∂Ψ) + Nα (Ψ, W ~ )∂α Ψ + N2 (Ψ, W ~ ), g(Ψ) Ψ = M(Ψ, W

(1.6.2a)

f e α (Ψ, W e 2 (Ψ, W ~ )∂α ∂β w = M(Ψ, ~ )Qg (∂Ψ, ∂Ψ) + N ~ )∂α Ψ + N ~ ). (h−1 )αβ (Ψ, W W 1

(1.6.2b)

1

Above and throughout, g and h are, by assumption, Lorentzian metrics for small values of their arguments (see below for our precise assumptions), g(Ψ) is the covariant wave e 2 are nonlinear terms described below, and operator29 of g(Ψ), M, Nα1 , · · · , and N Qg (∂Ψ, ∂Ψ) := (g −1 )αβ (Ψ)∂α Ψ∂β Ψ

(1.6.3)

is the standard null form associated to g. It is important for our proof that the wave operator of the shock-forming variable Ψ is covariant, the reason being that the geometric vectorfields that we construct exhibit good commutation properties with30 the operator g . We stress that Qg (∂Ψ, ∂Ψ) is, from the point of view of closing our estimates, the only allowable term on RHSs (1.6.2a)-(1.6.2b) that is quadratic in ∂Ψ. The reason is that Qg (∂Ψ, ∂Ψ) has a special nonlinear structure in that it is linear in the tensorial component of ∂Ψ that blows up; see (2.12.4) for the geo-analytic statement of this fact. More precisely, upon decomposing Qg (∂Ψ, ∂Ψ) relative to an appropriate frame, one find that it is linear in a derivative of Ψ in a direction transversal to the characteristics. The key point is that it is precisely the transversal derivative of Ψ that blows up, while the derivatives of Ψ in directions tangential to the characteristics remain uniformly bounded31 all the way up to the singularity. For this reason, such terms have only a negligible effect on the dynamics all the way up to the shock, at least compared to the Riccati-type term that is quadratic in the transversal derivatives of Ψ and that drives the singularity formation. Note that this Riccati-type term becomes visible only if we expand the expression g(Ψ) Ψ on LHS (1.6.2a) relative to Cartesian coordinates. We refer readers to [17] for further discussion on these issues, noting only that the good structure of Qg (∂Ψ, ∂Ψ) is referred to as the strong null condition (relative to g) in [17]. Note that typical inhomogeneous terms that are quadratic or higher-order in ∂Ψ are at least quadratic in the derivatives of Ψ in directions transversal to the characteristics. Such terms are too singular to be included on RHSs (1.6.2a)-(1.6.2b) within our framework and in 29Relative

to arbitrary coordinates, g f = p

1

∂α

p

|detg|(g −1 )αβ ∂β f .

|detg| precisely, they exhibit good commutation properties with µg , where we define µ in Def. 1.2. 31Except possibly at the high derivative levels, due to the degenerate energy estimates that we derive; see Subsubsect. 1.7.6. 30More

Shock formation for quasilinear wave systems featuring multiple speeds 18 fact, might introduce instabilities that prevent a shock from forming or, alternatively, that generate a completely different kind of blowup. In particular, like all prior works on shock formation, our proof is unstable against the addition of cubic terms (∂Ψ)3 to the equations, and similarly for terms that are higher-order in ∂Ψ. Remark 1.5 (Extending the result to a different type of fast wave equation). Instead of studying equation (1.6.2a), we could alternatively prove a shock-formation result for “fast” non-covariant quasilinear wave equations of the form (g −1 )αβ (∂Φ)∂α ∂β Φ = N(w, Φ, ∂Φ),

(1.6.4)

∂N (0, 0, 0) = 0 ∂(∂ν Φ) for ν = 0, 1, 2. As is explained in [27], the reason is that one could differentiate equation32 (1.6.4) with the Cartesian coordinate partial derivatives ∂ν to obtain a system of type ~ and W ~ that obey the inhomogeneous term assump(1.6.2a)-(1.6.2b) in the unknowns Φ, Ψ, ~ := (Ψ0 , Ψ1 , Ψ2 ) := (∂0 Φ, ∂1 Φ, ∂2 Φ) and g = g(Ψ). ~ tions described below, where Ψ More precisely it was shown in [27] that the scalar functions Ψν satisfy a system of covariant wave ~ exhibit the same kind equations of type (1.6.2a), where the terms that are quadratic in ∂ Ψ of good null structure as the standard g-null form (1.6.3).

where N(·) is a smooth function of its arguments such that N(0, 0, 0) =

1.6.2. Assumptions on the remaining nonlinearities. We assume that relative to the Cartee 2 (·) in the sian coordinates, the nonlinearities gαβ (·), hαβ (·), M(·), M1 (·), Nα1 (·), · · · , N ~| system (1.6.2a)-(1.6.2b) are given smooth functions of their arguments (for |Ψ| and |W sufficiently small) and that e α (0, 0) = N e 2 (0, 0) = 0, Nα1 (0, 0) = N2 (0, 0) = N 1

(α = 0, 1, 2).

(1.6.5)

In particular, under the assumptions (1.6.5), RHSs (1.6.2a)-(1.6.2b) are allowed to contain terms of the following form: order-unity terms times Ψ or Ψ∂Ψ. Such terms have not been treated in prior works on shock formation. Compared to terms treated in prior works, their new feature is that their derivatives in directions transversal to the characteristics of g(Ψ) are allowed to be large, even though, for the solutions under study here, the terms themselves are small in L∞ . The largeness of their transversal derivatives necessitates a different approach to deriving estimates for the higher transversal derivatives of Ψ (see Sect. 6), although such terms do not force us to alter our approach to energy estimates for Ψ; as we explain in Subsubsect. 1.7.6, our energy estimates are based on commuting the equations only with vectorfields that are tangential to the characteristics of g(Ψ), and for the solutions under study, the terms Ψ and Ψ∂Ψ exhibit smallness of the type considered in prior works. Note in particular that a Klein-Gordon-type inhomogeneous term (equal to a multiple of Ψ) does not prevent shock formation in the solution regime under consideration. Regarding the fast wave metric g, we assume that (Small)

gαβ = gαβ (Ψ) := mαβ + gαβ 32More

(1.6.4).

(Ψ),

(α, β = 0, 1, 2),

(1.6.6)

generally, our approach could be extended to allow for (g −1 )αβ = (g −1 )αβ (Φ, ∂Φ) in equation

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19

where mαβ = diag(−1, 1, 1) is the standard Minkowski metric on R × Σ (where Σ is defined (Small) in (1.1.1)) and the Cartesian components gαβ (Ψ) are given smooth functions of Ψ such that (Small)

gαβ

(Ψ = 0) = 0.

(1.6.7)

We also introduce the scalar functions d2 d gαβ (Ψ), G0αβ = G0αβ (Ψ) := gαβ (Ψ), (1.6.8) Gαβ = Gαβ (Ψ) := dΨ dΨ2 which appear throughout our analysis. In order to produce solutions to form shocks, we assume that Gαβ (Ψ = 0)Lα(F lat) Lβ(F lat) 6= 0,

(1.6.9)

L(F lat) = ∂t + ∂1 .

(1.6.10)

where

As is explained in [28], these assumptions are essentially equivalent to the assumption that the null condition fails for plane symmetric solutions in the wave equation for Ψ. Roughly, these assumptions ensure that for the solutions under study, the coefficient of the main terms driving the blowup is non-zero; as will become clear, the main term is the first product on RHS (2.10.1). Remark 1.6 (Genuinely nonlinear systems). Our assumption that the vectorfield (1.6.10) verifies (1.6.9) is similar the well-known genuine nonlinearity condition for first-order strictly hyperbolic systems. In particular, for plane symmetric solutions with Ψ sufficiently small, the assumption (1.6.9) ensures that there are quadratic Riccati-type terms in the fast wave equation (1.6.2a), which become visible if one expands the LHS relative to the Cartesian coordinates. The Riccati-type terms provide essentially the same mechanism that drives the singularity formation in the 2 × 2 genuinely nonlinear strictly hyperbolic systems studied by Lax [15]. As we mentioned above, a fundamental aspect of our proof is that we reformulate the slow wave equation (1.6.2b) as a first-order system, which which allows us to avoid certain top-order commutator error terms that we would have no means to control. Specifically, we study the following first-order system (which is easily seen to be a consequence of (1.6.2b)), (i = 1, 2): ~ )∂a wb + 2(h−1 )0a (Ψ, W ~ )∂a w0 ∂t w0 = (h−1 )ab (Ψ, W

(1.6.11a)

f e α (Ψ, W e 2 (Ψ, W ~ )Qg (∂Ψ, ∂Ψ) − N ~ )∂α Ψ − N ~ ), − M(Ψ, W 1 ∂t wi = ∂i w0 ,

(1.6.11b)

∂t w = w 0 ,

(1.6.11c)

∂i wj = ∂j wi .

(1.6.11d)

Note that (1.6.11d) can be viewed as a constraint representing the symmetry of the mixed partial derivatives of w with respect to the Cartesian coordinates. It is easy to check that the constraint (1.6.11d), if verified at time 0, is propagated by the flow of equation (1.6.11b).

Shock formation for quasilinear wave systems featuring multiple speeds 20 1.6.3. Assumptions tied to the wave speeds. We now quantify our assumption that Ψ is the fast wave and w is the slow wave. Assumption on the wave speeds ~ | is sufficiently We assume that the following holds for non-zero vectors V whenever |Ψ|+|W small: hαβ V α V β ≤ 0 =⇒ gαβ V α V β < 0.

(1.6.12)

Note that (1.6.12) is equivalent to the following implication, valid for non-zero co-vectors ω: (g −1 )αβ ωα ωβ ≤ 0 =⇒ (h−1 )αβ ωα ωβ < 0.

(1.6.13)

From (1.6.13) and the fact that (g −1 )αβ (Ψ = 0) = (m−1 )αβ = diag(−1, 1, 1) (see (1.6.6)~ ) < 0 whenever |Ψ| and |W ~ | are sufficiently small. For (1.6.7)), it follows that (h−1 )00 (Ψ, W convenience, we rescale the metrics and equations by a positive conformal factor so that the following holds relative to the Cartesian coordinates: ~ ) ≡ −1. (g −1 )00 (Ψ) = (h−1 )00 (Ψ, W (1.6.14) The identities assumed in (1.6.14) simplify many calculations but are in no way essential. Note that in view of the definition of a covariant wave operator, rescaling the metric g introduces an additional inhomogeneous null form term of the form M(Ψ)Qg (∂Ψ, ∂Ψ) on RHS (1.6.2a). It turns out that this term is not important for the dynamics due to its good null structure. Note also that this new term already falls under the scope of the allowable terms on RHS (1.6.2a). 1.7. Overview of the proof of the main result. In this subsection, we overview of the proof of our main result, Theorem 10.1. Our basic geometric setup is similar to the one pioneered by Christodoulou in his study of shock formation in irrotational relativistic fluid mechanics [4]. 1.7.1. Basic geometric ingredients. As in all prior works on shock formation in more than one spatial dimension, to follow the solution all the way to the singularity in ∂Ψ, we construct an eikonal function adapted to the metric g(Ψ). Definition 1.1 (Eikonal function). The eikonal function u solves the eikonal equation initial value problem (g −1 )αβ (Ψ)∂α u∂β u = 0,

∂t u > 0, 1

u|Σ0 = 1 − x ,

(1.7.1a) (1.7.1b)

where Σ0 ' R × T is the hypersurface of constant Cartesian time 0. Our choice of initial conditions in (1.7.1b) is adapted to the approximate plane symmetry of the data that we will consider. The level sets of u are g-null hypersurfaces, which we denote by Pu (see Def. 1.3) and which we often refer to as the characteristics. See Figure 1 for a depiction of the characteristics, where the characteristics Put in the figure have been truncated at time t. We clarify that even though the system (1.6.2a) + (1.6.11a)-(1.6.11d) features multiple speeds of propagation, we study only the characteristic family {Pu }u∈[0,1]

J. Speck

21

in detail since, for the data under consideration, the intersection of distinct members of this family corresponds to the formation of a shock. Using u, we will construct a collection of geometric objects that can be used to derive sharp information about the solution. The most important of these is the inverse foliation density µ. Its vanishing corresponds to the intersection of the characteristics and, as it turns out (see Subsubsect. 1.7.5), the formation of a singularity in ∂α Ψ. Definition 1.2 (Inverse foliation density). We define µ > 0 as follows: −1 µ := −1 αβ , (g ) (Ψ)∂α t∂β u

(1.7.2)

where t is the Cartesian time coordinate. Note that by (1.6.6)-(1.6.7) and the initial conditions (1.7.1b) for u, we have µ|Σ0 = 1 + O(Ψ). Thus, for the data considered in this article in which |Ψ| is small, it follows that µ is initially near unity. In short, our main goal in the article is to exhibit an open set of data such that µ vanishes in finite time and to show that its vanishing is tied to the blowup of ∂α Ψ. The following spacetime subsets are tied to u and play a fundamental role in our analysis. Definition 1.3 (Subsets of spacetime). We define the following subsets of spacetime: Σt0 := {(t, x1 , x2 ) ∈ R × R × T | t = t0 }, u0 t0

(1.7.3a)

Σ := {(t, x1 , x2 ) ∈ R × R × T | t = t0 , 0 ≤ u(t, x1 , x2 ) ≤ u0 },

(1.7.3b)

Pu0 := {(t, x1 , x2 ) ∈ R × R × T | u(t, x1 , x2 ) = u0 },

(1.7.3c)

t0 u0

P := {(t, x1 , x2 ) ∈ R × R × T | 0 ≤ t ≤ t0 , u(t, x1 , x2 ) = u0 }, t0 u0

u0 t0

`t0 ,u0 := P ∩ Σ = {(t, x1 , x2 ) ∈ R × R × T | t = t0 , u(t, x1 , x2 ) = u0 }, t0

Mt0 ,u0 := ∪u∈[0,u0 ] Pu ∩ {(t, x1 , x2 ) ∈ R × R × T | 0 ≤ t < t0 }.

(1.7.3d) (1.7.3e) (1.7.3f)

We refer to the Σt and Σut as “constant time slices,” the Put as “characteristics” or “null hypersurfaces,” and the `t,u as “tori.” Note that Mt,u is “open-at-the-top” by construction. To study the solution, we complement t and u with a geometric torus coordinate ϑ to form a geometric coordinate system (t, u, ϑ) with corresponding partial derivative vectorfields ∂ ∂ ∂ , , Θ := . To differentiate the equations and obtain estimates for the solution’s ∂t ∂u ∂ϑ derivatives, we also construct a related vectorfield frame ˘ Y }, {L, X, (1.7.4) ∂ which spans the tangent space at each point where µ > 0. The vectorfield L verifies L = ∂t ∂ ˘ and is null with respect to g, while X and Y are, respectively, replacements for and Θ ∂u with better regularity properties that are needed to close the top-order energy estimates; see Subsect. 2.1 for the details behind the construction of ϑ and the vectorfields. We will prove that for the solutions under study, the vectorfields L and Y remain close to their background ˘ behaves like −µ∂1 and thus values, which are respectively ∂t + ∂1 and ∂2 . In contrast, X

Shock formation for quasilinear wave systems featuring multiple speeds 22 1 ˘ X remains close to −∂1 all the way up to µ ˘ Y } and note in the shock. See Figure 1 for a schematic depiction of the vectorfields {L, X, ˘ is smaller in the region where µ is small. Note also that we have displayed particular that X the characteristics of g(Ψ) in the figure but not those of h since they do not play a role in the ˘ is transversal analysis. Moreover, L and Y are tangential to the characteristics Pu while X to them. A key aspect of our proof is that we will be able to derive uniform bounds for ˘ derivatives of the solution all the way up to the shock, except near the the L, Y , and X top derivative level; as we describe in the discussion surrounding (1.7.33a)-(1.7.33g), our high-order energies are allowed to blow up as the shock forms. The fact that we can derive ˘ derivatives of the solution is fundamentally tied to non-singular estimates for the low-level X ˘ shrinks like µ as µ → 0 and is compatible with the formation of a singularity the fact that X in the Cartesian coordinate partial derivatives ∂α Ψ. More precisely, our main theorem yields ˘ & 1 near points where µ is small and thus the derivative of Ψ with respect to the that |XΨ| 1 ˘ order-unity vectorfield X := X blows up precisely when µ vanishes; see Subsubsect. 1.7.5 µ for a more detailed overview of this aspect of the proof. shrinks as the shock forms. Moreover, X :=

µ small

L Y ˘ X L

Y

µ≈1 ˘ X P1t

Put

P0t

Figure 1. The vectorfield frame from (1.7.4) at two distinct points in Pu 1.7.2. The spacetime regions under study. For convenience, we study only the future portion of the solution that is completely determined by the data lying in the subset ΣU0 0 ⊂ Σ0 of thickness U0 and on a portion of the characteristic P0 , where 0 < U0 ≤ 1

(1.7.5)

is a parameter, fixed until Theorem 10.1; see Figure 2. We will study spacetime regions such that 0 ≤ u ≤ U0 , where u is the eikonal function from Def. 1.1. We have introduced the

J. Speck

23

ve r

y

sm al l

da ta

parameter U0 because one would need to allow U0 to vary in order to study the behavior of the solution up the boundary of the maximal development, as Christodoulou did in [4, Chapter 15]. For brevity, we do not pursue this issue in the present article.

PUt 0

x2 ∈ T

P0t

ΣU0 0 U0 “interesting” data x1 ∈ R

Figure 2. The spacetime region under study. In our analysis, we will restrict our attention to times t verifying 0 ≤ t < 2˚ δ−1 ∗ , where ˚ δ∗ > 0 is the data-dependent parameter defined by h i 1 ˚ ˘ δ∗ := sup GLL XΨ . (1.7.6) 2 Σ10 − The quantity (1.7.6) is essentially the main term in the transport equation for µ (see (2.10.1)) that drives µ to 0 in finite time. In (1.7.6), GLL := Gαβ Lα Lβ , where Gαβ is defined in (1.6.8) and L is the g-null vectorfield mentioned in Subsubsect. 1.7.1 (see Def. 2.3 for the precise definition). In our analysis, we take into account only the portion of the data lying in the ˚−1 subset P02δ∗ of the characteristic P0 since, by domain of dependence considerations, only this portion can influence the solution in the regions under study. The parameter ˚ δ∗ is important because under certain assumptions described below, the time of first shock formation is a ˚ small perturbation33 of ˚ δ−1 ∗ . We will clarify the connection between δ∗ and the time of first shock formation in Subsubsect. 1.7.5. Moreover, in view of the above remarks, we see that to close our bootstrap argument (which we briefly overview in Subsubsect. 1.7.4), it is sufficient to control the solution for times up to 2˚ δ−1 ∗ , which is plenty of time for the shock to form. 1.7.3. A model problem: shock formation for plane symmetric, nearly simple waves. In this subsubsection, we illustrate some of the main ideas behind our analysis by sketching a proof of our main results for plane symmetric solutions, that is, solutions that depend only on t ˚ sufficiently small, the time of first shock formation is {1 + O(˚ )}˚ δ−1 is the data-size ∗ , where ˚ parameter described in Subsubsect. 1.7.4. 33For

Shock formation for quasilinear wave systems featuring multiple speeds 24 and x1 . For such solutions, we are able to rely exclusively on the method of characteristics when deriving estimates. In particular, we can avoid energy estimates, which drastically simplifies the proof. Our analysis in this subsubsection can be viewed as a sharpening of the approach of John [11], in the spirit of the recent work [8]. For convenience, we consider only the case in which the fast wave metric perturbation function from (1.6.6) takes the simple form (Small) gαβ (Ψ) = (1 + Ψ)2 − 1 δα1 δβ1 , (1.7.7) where δαβ is the standard Kronecker delta. Moreover, in this subsubsection only, we use, in ˘ defined by addition to the vectorfield L, the vectorfield L ˘ := µL + 2X. ˘ L (1.7.8) ˘ L) ˘ = 0 (that is, that L ˘ is g-null) and that the following relations It is easy to check that g(L, hold (these relations follow easily from Lemma 2.1): Lt = 1,

˘ = µ, Lt

Lu = 0,

˘ = 2. Lu

(1.7.9)

˘ plays From the point of view of the estimates derived in this subsubsection, the vectorfield L ˘ that we use in the rest of the paper. The a similar role to the Pu -transversal vectorfield X ˘ advantage of L in this subsubsection is that it is g-null and thus the principal part of the fast wave equation takes a simple form in plane symmetry when expressed in terms of L and ˘ derivatives; see equations (1.7.13a)-(1.7.13b) L As in the bulk of the paper, we will focus our attention here on nearly simple waves. By an outgoing (that is, right-moving) simple plane wave, we mean a solution such that LΨ ≡ 0 and w ≡ 0. Due to the inhomogeneous terms on RHS (1.6.11a), the systems under study here do not generally admit simple plane wave solutions. However, they do admit approximately ˘ and simple outgoing plane wave solutions such that Ψ and LΨ are small relative to LΨ ˘ are small relative to LΨ ˘ as well. In the present subsubsection, we such that w, Lw and Lw will consider plane symmetric initial data verifying such size assumptions. Our assumptions involve the parameters ˚ , ˚ δ, and ˚ δ∗ , which in this subsubsection only have slightly different (but analogous) definitions than they do in the rest of the paper. Specifically, we assume that the initial data for Ψ and w are given along Σ10 , which corresponds to the portion of Σ0 with ˚−1 0 ≤ u ≤ 1, as well as P02δ∗ , which is the portion of the level set {u = 0} with 0 ≤ t ≤ 2˚ δ−1 ∗ , 1 ˚ where we define δ∗ just below. Note that in plane symmetry, Σ0 can be identified with the unit interval [0, 1] of x1 values. We assume the following size conditions, where all functions on the LHSs of the inequalities are assumed to be continuous with respect to the geometric coordinates (t, u): ˘ Σ1 = f (u) + O(˚ LΨ| ), 0 kΨkL∞ (Σ1 ) , kLΨkL∞ (Σ1 ) ≤ ˚ ,

(1.7.10b)

kwkL∞ (Σ1 ) , kw0 kL∞ (Σ1 ) , kw1 kL∞ (Σ1 ) ≤ ˚ ,

(1.7.10c)

0

0

2˚ δ−1 ∗ )

L∞ (P0

2˚ δ−1 ∗ )

L∞ (P0

0

0

kΨk kwk

(1.7.10a)

, kw0 k

, kLΨk

2˚ δ−1 ∗ )

L∞ (P0

, kw1 k

0

2˚ δ−1 ∗ )

L∞ (P0

2˚ δ−1 ∗ )

L∞ (P0

≤˚ ,

(1.7.10d)

≤˚ ,

(1.7.10e)

J. Speck

25

where f (u) is a continuous function and ˚ δ∗ = sup [f (u)]− .

(1.7.11)

u∈[0,1]

Above, ˚ is a parameter that is small relative to ˚ δ∗ and small relative to ˚ δ−1 , where ˚ δ := sup |f (u)|.

(1.7.12)

u∈[0,1]

For convenience, in this subsubsection, we study only the following specific example of a system of type (1.6.2a) + (1.6.11a)-(1.6.11d) in one spatial dimension, where the metric perturbation function is given by (1.7.7) and we have ignored certain harmless interaction terms in order to simplify the discussion: ˘ = LΨ · LΨ ˘ + w0 · LΨ ˘ + µw0 · Ψ, LLΨ ˘ ˘ + µ(LΨ)2 + w0 · LΨ ˘ + µw0 · Ψ, LLΨ = LΨ · LΨ 1 ˘ + µw0 · Ψ, µ∂t w0 = µ∂1 w1 + LΨ · LΨ 4 µ∂t w1 = µ∂1 w0 .

(1.7.13a) (1.7.13b) (1.7.13c) (1.7.13d)

We now make some remarks on the structure of equations (1.7.13a)-(1.7.13d). We have multiplied the equations by the inverse foliation density µ, which will help clarify certain aspects of the analysis.34 The forms of LHSs (1.7.13a)-(1.7.13b) are a consequence of Prop. 2.12. In 1 (1.7.13c), the factor of accounts for our assumption that w is the slow wave. Note that 4 equations (1.7.13c)-(1.7.13d) are semilinear while for the general class of equations that we consider, the analogous equations are typically quasilinear. These facts play very little role in the discussion in this subsubsection. In particular, in proving the main theorem of the paper, we use that w is the slow wave mainly when deriving energy estimates, which we can avoid in this subsubsection by integrating along characteristics. To facilitate our analysis via integrating along characteristics, we now replace (1.7.13c)(1.7.13d) with the following equations,35 which are equivalent up to harmless constant factors on the right-hand sides: 1 ˘ + µw0 · Ψ, µ(2∂t + ∂1 )(w0 − w1 ) = LΨ · LΨ (1.7.14a) 2 1 ˘ + µw0 · Ψ. µ(2∂t − ∂1 )(w0 + w1 ) = LΨ · LΨ (1.7.14b) 2 To proceed, we rely on the following relations, which are simple consequences of Lemma 2.13, our assumption that the metric perturbation is given by (1.7.7), the normalization condition g(X, X) = 1 (see (2.2.10a)), our assumption of plane symmetry, and the fact that (under 34Away

from plane symmetry, it is critically important to multiply the wave equations by µ before commuting them with appropriate vectorfields; the factor of µ leads to important cancellations. In contrast, our arguments in this subsubsection do not involve commuting the equations. 35Equations (1.7.14a)-(1.7.14b) are evolution equations for the Riemann invariants of the subsystem (1.7.13c)-(1.7.13d).

Shock formation for quasilinear wave systems featuring multiple speeds 26 these assumptions) Y = ∂2 : 1˘ 2∂t = L + L, µ

1˘ 2∂1 = (1 + Ψ) L − L . µ

(1.7.15)

Using (1.7.15), we can replace (1.7.14a)-(1.7.14b) with the following equations, which are again equivalent to (1.7.13c)-(1.7.13d) up to harmless constant factors on the right-hand sides: n o ˘ + µ(3 + Ψ)L (w0 − 1 w1 ) = LΨ · LΨ ˘ + µw0 · Ψ, (1 − Ψ)L (1.7.16a) 2 n o 1 ˘ ˘ + µw0 · Ψ. (3 + Ψ)L + µ(1 − Ψ)L (w0 + w1 ) = LΨ · LΨ (1.7.16b) 2 In plane symmetry, the most important aspect of LHSs (1.7.16a)-(1.7.16b) are that for |Ψ| ˘ + µ(3 + Ψ)L and (3 + Ψ)L ˘ + µ(1 − Ψ)L are transversal to small, the vectorfields (1 − Ψ)L ˘ = 2 and Lu = 0 (see (1.7.9)). the Pu , a simple fact that follows from the identities Lu We now note that µ (which is defined in (1.7.2)) verifies an evolution equation that we can schematically express as follows (see (2.10.1) for the precise formula): ˘ + µLΨ. Lµ = LΨ

(1.7.17)

In total, we will study the system (1.7.13a)-(1.7.13b) + (1.7.16a)-(1.7.16b) + (1.7.17) and sketch a proof that whenever ˚ is sufficiently small (in a manner that is allowed to depend ˚ ˚ on δ and δ∗ ), a shock forms in Ψ in finite time. To facilitate our analysis, it is convenient to make the following bootstrap assumptions for (t, u) ∈ [0, T(Boot) ) × [0, 1], where 0 < T(Boot) ≤ 2˚ δ−1 ∗ is a bootstrap time: √ |Ψ|, |LΨ|, |w0 |, |w1 | ≤ ˚ , (1.7.18a) √ ˘ , (1.7.18b) LΨ(t, u) − f (u) ≤ ˚ √ |µ(t, u)| ≤ 1 + 2|f (u)|˚ δ−1 . (1.7.18c) ∗ + ˚ We also assume that for (t, u) ∈ [0, T(Boot) ) × [0, 1], we have µ(t, u) > 0,

(1.7.19)

which is tantamount to the assumption that a shock has not yet formed on [0, T(Boot) ) × [0, 1], though it allows for the possibility that a shock forms exactly at time T(Boot) . By standard local well-posedness, if the data verify the size assumptions (1.7.10a)-(1.7.10e), ˚ is sufficiently small, and T(Boot) is sufficiently small, then there exists a classical solution for (t, u) ∈ [0, T(Boot) ) × [0, 1] on which the bootstrap assumptions are verified. Using the identities (1.7.15), we see that if the bootstrap assumptions are not saturated and µ remains uniformly positive on [0, T(Boot) )×[0, 1], then the solution and its ∂t and ∂1 derivatives remain uniformly bounded in magnitude on [0, T(Boot) ) × [0, 1]. It is a standard result that under these conditions, the solution can be classically continued past the time T(Boot) . Thus, in order prove shock formation, it suffices to improve the bootstrap assumptions by showing √ replace by C˚ (for ˚ sufficiently small), to show that µ can vanish in that they hold with ˚ finite time, and to show that its vanishing leads to the blowup of some first-order Cartesian coordinate partial derivative of Ψ.

J. Speck

27

We now explain how to improve the bootstrap assumptions, starting with (1.7.18a). To this end, it is convenient to introduce q(u) := sup {|LΨ| + |w0 | + |w1 |} .

(1.7.20)

T(Boot)

Pu

d In the rest of the proof, we implicitly rely on (1.7.9), which allows us to think of L = dt d ˘ =2 ˘ Similarly, we along the integral curves of L and L along the integral curves of L. du ˘ + µ(3 + Ψ)L = 2(1 − Ψ) d along the integral curves of L ˘ + 3µL and have that (1 − Ψ)L du ˘ + µ(1 − Ψ)L = 2(3 + Ψ) d along the integral curves of (3 + Ψ)L ˘ + µ(1 − Ψ)L. Using (3 + Ψ)L du these observations, we integrate equations (1.7.13b) and (1.7.16a)-(1.7.16b) and use (1.7.9), the bootstrap assumptions, and the small-data assumptions (1.7.10b), (1.7.10c), (1.7.10d), and (1.7.10e) to obtain Z u q(u0 ) du0 , (1.7.21) q(u) ≤ C˚ +C u0 =0

where here and throughout the paper, all constants C are allowed to depend on ˚ δ and ˚ δ∗ , and similarly for implicit constants hidden in the notation . and O. From (1.7.21) and Gronwall’s inequality, we conclude that supu∈[0,1] q(u) . ˚ . Next, using the already obtained bound |LΨ| . ε, the fundamental theorem of calculus, and the data-size assumptions (1.7.10b) for Ψ, we deduce that for (t, u) ∈ [0, T(Boot) ) × [0, 1], we have Z t |LΨ|(s, u) ds (1.7.22) |Ψ| (t, u) ≤ C˚ + ≤ C˚ +

s=0 C˚ δ−1 ≤ ∗ ˚

C˚ .

We have thus derived the desired improvements of the bootstrap assumptions (1.7.18a). Next, using the previously obtained estimates, the bootstrap assumptions, equation (1.7.13a), the fundamental theorem of calculus, and the data assumption (1.7.10a), we obtain Z t ˘ ˚ +C ds (1.7.23) LΨ(t, u) − f (u) ≤ C˚ s=0

≤ C˚ + C˚ δ−1 ≤ C˚ , ∗ ˚ which yields an improvement of the bootstrap assumption (1.7.18b). Similarly, from the previously obtained estimates, the bootstrap assumptions, equation (1.7.17), and the fact that (by construction) µ|t=0 = 1 + O(Ψ), we deduce that µ(t, u) = 1 + f (u)t + O(˚ ).

(1.7.24)

We have therefore improved the bootstrap assumption (1.7.18c), which completes our proof of the improvement of the bootstrap assumptions. We now show that a shock forms in finite time. We start by setting µ? (t) := min µ(t, u). u∈[0,1]

Shock formation for quasilinear wave systems featuring multiple speeds 28 From (1.7.11) and (1.7.24), we find that µ? (t) = 1 − ˚ δ∗ t + O(˚ ).

(1.7.25)

From (1.7.25), we easily infer that µ? (t) vanishes at the time T(Shock) = {1 + O(˚ )} ˚ δ−1 ∗ . Finally, from (1.7.11), (1.7.15), the bounds |Ψ| . ˚ and |LΨ| . ˚ , and (1.7.23), we see that supu∈[0,1] |∂t Ψ(t, u)| and supu∈[0,1] |∂1 Ψ(t, u)| are equal to non-zero, bounded functions times 1 as t ↑ T(Shock) and thus blow up precisely at time T(Shock) . µ? 1.7.4. The full problem without symmetry assumptions: data-size assumptions, bootstrap assumptions, and L∞ estimates. In our main theorem (Theorem 10.1), we study (nonsymmetric) perturbations of the plane symmetric nearly simple outgoing wave solutions studied in Subsubsect. 1.7.3. We now outline the size assumptions that we make on the data in proving our main theorem. Our assumptions are similar in spirit to our data assumptions from Subsubsect. 1.7.3 but are more complicated in view of the additional spatial direction and the necessity of deriving energy estimates away from plane symmetry; see Subsect. 3.3 for a precise statement of our assumptions. We study solutions such that the interesting, ˚−1 relatively large portion of the data lies in ΣU0 0 while the data on P02δ∗ are very small; see Figure 2 on pg. 23. Here and in the remainder of the article, ˚ δ∗ > 0 denotes the data-dependent parameter defined in (1.7.6). We consider data for Ψ such that along ΣU0 0 , Ψ itself and its Pu -tangential derivatives up to top-order (that is, its L and Y derivatives) are initially of ˘ derivatives such as XΨ ˘ and a relatively small size ˚ in appropriate norms while its pure X 36 ˚ ˘ ˘ X XΨ are of a relatively large size δ. We assume that mixed tangential derivatives such as ˘ are also of a relatively small size ˚ LXΨ . Our size assumptions are such that the energies we use to control the solution are all initially of small size O(˚ 2 ); see Subsubsect. 1.7.6 for further discussion on this point. These size assumptions are the same as the ones made in [16, 28] and correspond to data close to that of the nearly simple outgoing plane waves studied in Subsubsect. 1.7.3. Roughly, the relative largeness of ˚ δ is tied to a Riccati-type blowup of the first Cartesian coordinate partial derivative vectorfields of Ψ. We assume that ~ up to top-order are initially of small size all derivatives of the slow wave variable array W 37 ˘ ˘ ~ ˚ , except for X X W , which we allow to be of relatively large size ˚ δ. Finally, we assume 2˚ δ−1 ∗ ~ that along P0 , all derivatives of Ψ and W up to top-order are of small size ˚ . See the beginning of Sect. 3 for some remarks on the existence of such data, which is compatible38 with the structure of the nonlinearities in the system (1.6.2a) + (1.6.11a)-(1.6.11d). The data-size assumptions described in the previous paragraph correspond to a pair of ~) waves in which one wave (namely Ψ) is nearly simple and outgoing while the other (W is uniformly small. A key point of our proof is showing how to propagate various aspects of this ˚ -˚ δ hierarchy all the way up to the shock, much like in in Subsubsect. 1.7.3. As in Subsubsect. 1.7.3, to propagate the ˚ -˚ δ hierarchy, it is convenient to make L∞ bootstrap ~ , and their geometric derivatives on a bootstrap time interval of the assumptions for Ψ, W 36˚ δ is

allowed to be small in an absolute sense. ˘ derivatives of W ~. our proof, we do not need estimates for more than two X 38By “compatible,” we mean that one needs to differentiate the equations to obtain estimates for some of the higher derivatives of the data. 37In

J. Speck

29

form [0, T(Boot) ), on which µ > 0 and on which the solution exists classically. In view of the remarks made below (1.7.6), we can assume that T(Boot) ≤ 2˚ δ−1 ∗ . The assumptions are (see Subsect. 3.6)

≤11 ~

P Ψ ∞ u , ≤ ε, (1.7.26)

P ≤10 W

L (Σt )

L∞ (Σu t)

where P ≤M denotes an arbitrary order ≤ M differential operator corresponding to repeated differentiation with respect to the Pu -tangential vectorfields P = {L, Y } and ε is a small bootstrap parameter that at the end of the paper, by virtue of a priori energy estimates and Sobolev embedding, will have been shown to verify ε . ˚ . Remark 1.7. Note that the uniform boundedness of ∂α w up to the shock is already accounted for in the bootstrap assumption (1.7.26). Note furthermore that the same is not true for ∂α Ψ, which blows up at the shock. Using (1.7.26) and our data-size assumptions, we can derive L∞ estimates for the low-order ~ on the bootstrap pure transversal and mixed transversal-tangential derivatives of Ψ and W i region and for various derivatives of µ and L for times up as large as 2˚ δ−1 ∗ . Roughly, this is the content of Sects. 5 and 6. The analysis is similar in spirit to that of Subsubsect. 1.7.3 but is much more involved. We need the estimates for the derivatives of µ and Li because these quantities arise as error terms when we commute the equations with the vectorfields ˘ Y }. {L, X, 1.7.5. Proof sketch of the formation of the shock and the blowup of ∂Ψ. Given the L∞ estimates described in Subsubsect. 1.7.4, the proof that µ → 0 in finite time and that some Cartesian coordinate partial derivative of Ψ blows up are not much more difficult they were in Subsubsect. 1.7.3. We now sketch the proof. First, one derives (essentially as a consequence of the eikonal equation (1.7.1a)) the following transport equation for µ (see Lemma 2.8): 1 ˘ Lµ(t, u, ϑ) = [GLL XΨ](t, u, ϑ) + µO(P Ψ)(t, u, ϑ). (1.7.27) 2 In (1.7.27) and throughout, P schematically denotes a differentiation in a direction tangential ~. to the characteristics Pu . Note that equation (1.7.27) does not involve the slow wave W Using bootstrap assumptions and L∞ estimates of the type described in Subsubsect. 1.7.4, it ˘ ˘ is easy to show that µO(P Ψ)(t, u, ϑ) = O(ε) and that [GLL XΨ](t, u, ϑ) = [GLL XΨ](0, u, ϑ)+ O(ε). Inserting these estimates into (1.7.27), we find that 1 ˘ u, ϑ) + O(ε). (1.7.28) Lµ(t, u, ϑ) = [GLL XΨ](0, 2 From (1.7.28), definition (3.3.1) and the fact that ε is controlled by ˚ , we see that there exists (u∗ , ϑ∗ ) ∈ [0, 1] × T such that Lµ(t, u∗ , ϑ∗ ) = −˚ δ∗ + O(˚ ). (1.7.29) ∂ and that µ|t=0 = 1 + O(Ψ) = 1 + O(˚ ), we see that if ˚ is sufficiently ∂t small, then µ vanishes for the first term at {1 + O(˚ )}˚ δ−1 ∗ . Moreover, µ vanishes linearly in that Lµ is strictly negative at the vanishing points; as we described in Subsubsect. 1.7.6, these are crucially important facts for our energy estimates. Finally, we note that the above Recalling that L =

Shock formation for quasilinear wave systems featuring multiple speeds 30 ˘ & 1 at the points where µ vanishes and thus XΨ := 1 XΨ ˘ argument also yields that |XΨ| µ 1 blows up like at such points. Since we are also able to show that the Cartesian components µ X α remain close to −δ1α throughout the evolution, it follows that some Cartesian coordinate partial derivatives of Ψ must blow up when µ vanishes.

1.7.6. Overview of the energy estimates. Energy estimates are by far the most difficult aspect of the proof. For reasons to be explained, to close our energy estimates, we must commute the evolution equations up to 18 times with of elements the Pu -tangential commutation set P = {L, Y } and derive energy estimates for the differentiated quantities. The starting ~ obtained by applying the point for these energy estimates is energy identities for Ψ and W divergence theorem on the regions depicted in Figure 3. We provide the details behind these energy identities in Sect. 4; here we focus mainly on outlining how to derive a priori energy estimates based on the energy identities.

Σut `t,u `t,0

Put

x2 ∈ T

`0,u

Σu0

Mt,u

P0t

`0,0

x1 ∈ R

Figure 3. The energy estimate region

We start by describing our energy-flux quantities. In Subsubsect. 1.7.6 only, we denote ~ by H(Slow) . Schematically, the energy-flux quantity39 for Ψ by H(F ast) and the one for W (F ast) (Slow) H and H have the following strength (see Sect. 4 for precise statements concerning the energies and their coerciveness), where the integrals are with respect to the geometric

39In our detailed analysis, when constructing L2 -controlling quantities, we separately define energies along Σut ,

null fluxes along Put , and spacetime integrals over Mt,u . Here, to shorten our explanation of the main ideas, we have grouped them together.

J. Speck

31

coordinates:40 H

(F ast)

Z

Z n o 2 2 2 0 ˘ µ(LΨ) + (XΨ) + µ(Y Ψ) dϑdu +

(t, u) ∼ Σu t

(LΨ)2 + µ(Y Ψ)2 dϑdt0

Put

(1.7.30a) Z + Z (Slow) H (t, u) ∼

[Lµ]− (Y Ψ)2 dϑdu0 dt0 , Mt,u Z 2 0 ~ | dϑdu + ~ |2 dϑdt0 . µ|W |W

Σu t

(1.7.30b)

Put

On RHS (1.7.30a) and throughout, f− := max{−f, 0}. A crucially important feature of the above energies is that some of the integrals on RHSs (1.7.30a)-(1.7.30b) are µ-weighted and thus become weak near the shock, that is, in regions where µ is near 0. It turns out that the µ-weighted integrals are not able to suitably control all of the error terms that arise in the energy identities. The reason is that we encounter some error terms that lack µ weights and that are therefore relatively strong. However, it is ˘ Y Ψ, W ~ } are controlled by one of H(F ast) or H(Slow) also true that all elements of {LΨ, XΨ, without a µ weight and thus, in total, we are able to control all error integrals. Let us further Z comment on the coercivity of the spacetime integral [Lµ]− (Y Ψ)2 dϑdu0 dt0 featured on Mt,u

RHS (1.7.30a). The key point is that Lµ is quantitatively negative in regions where µ is small (and thus [Lµ]− is positive), which leads to the coerciveness of the integral. The reasons behind this were outlined in Subsubsect. 1.7.5. In all prior works on shock formation in more than one spatial dimension, similar spacetime integrals were exploited to close the energy estimates. The idea to exploit such a spacetime integral seems to have originated in the works [3, 4]. (F ast) We now let HN denote an energy corresponding to commuting the wave equation for Ψ with a string of vectorfields P N consisting of precisely N factors of elements the Pu (Slow) ~ . As tangential commutation set P = {L, Y }. We let HN be an analogous energy for W we alluded to above, in our detailed proof, we will have 1 ≤ N ≤ 18. For such N values, our initial data are such that all energies are of initially small size O(˚ 2 ), where ˚ is the smallness (F ast) parameter from Subsubsect. 1.7.4. In particular, we avoid using the energy H0 , which ˘ and is therefore allowed to be of involves the L2 norm of the pure transversal derivative XΨ a relatively large size O(˚ δ2 ); as we mentioned earlier, in order to close our proof, we do not ˘ in L2 , but rather only in L∞ and similarly for X ˘ XΨ ˘ and X ˘X ˘ XΨ, ˘ need to control XΨ for reasons that we clarify starting in Sect. 6. We can obtain these estimates by commuting the ˘ and treating it like a transport equation in wave equation (1.6.2a) up to two times with X ˘ with source terms that are controlled by our energies; see Props. 5.6 and 6.2 for detailed XΨ proofs. We now provide a few more details about how to derive the energy identities that form the starting point of our L2 -type analysis. To obtain the relevant energy identities for Ψ, we 40In

our schematic overview of the proof, we use the notation A ∼ B to imprecisely indicate that A is well-approximated by B.

Shock formation for quasilinear wave systems featuring multiple speeds 32 commute the wave equation (1.6.2a) with P N , multiply by T P N Ψ, where ˘ T := (1 + 2µ)L + 2X

(1.7.31)

is a “multiplier vectorfield” with appropriately chosen µ weights, and then integrate by parts ~ , we commute equations over Mt,u . Similarly, to obtain the relevant energy identities for W N (1.6.11a)-(1.6.11d) with P , multiply by an appropriate quantity, and then integrate by parts over Mt,u ; see Sect. 4 for the details behind the integration by parts and Sect. 8 for the details behind the pointwise estimates for the inhomogeneous terms on RHSs (1.6.2a) and (1.6.11a) and for the error terms that we generate upon commuting the equations. In total, we can use these energy identities and pointwise estimates to obtain a system of integral inequalities of the following type, where here we only schematically display a few representative terms: Z (F ast) 2 ˘ ˘ N Ψ)Y N trg/ χ dϑdu0 dt0 (t, u) ≤ C˚ + (XΨ)( XY (1.7.32a) HN Mt,u Z ~ dϑdu0 dt0 + · · · , (LY N Ψ)Y N W + Mt,u Z Z (Slow) N ~ 2 0 0 2 ~ )(XY ˘ N Ψ) dϑdu0 dt0 + · · · . (Y N W |Y W | dϑdu dt + HN (t, u) ≤ C˚ + Mt,u

Mt,u

(1.7.32b) The C˚ 2 terms on RHSs (1.7.32a)-(1.7.32b) are generated by the data. The tensorfield χ on RHS (1.7.32a) is the null second fundamental form of the co-dimension-two tori `t,u . It is a symmetric type 02 tensorfield with components χΘΘ = g(DΘ L, Θ) (see (2.5.1a) and ∂ recall that Θ = ), where D is the Levi-Civita connection of g. Moreover, trg/ χ is the trace ∂ϑ of χ with respect to the Riemannian metric g/ induced on `t,u by g. Geometrically, trg/ χ is the null mean curvature of the g-null hypersurfaces Pu . Analytically, we have Lα ∼ ∂u (see (2.2.4)) and thus trg/ χ ∼ ∂ 2 u, where u is the eikonal function. From the point of view of counting derivatives, one might expect to see terms such as Y N trg/ χ ∼ ∂ N +2 u on the RHS of the N -times commuted fast wave equation since the Cartesian components P α of the elements P ∈ P = {L, Y } depend on ∂u; roughly, terms such as Y N trg/ χ can arise when one commutes operators of the form P N through41 the expression g Ψ and the maximum number of derivatives falls on the components42 P α ∼ ∂u. Thus, the presence of Y N trg/ χ on RHS (1.7.32a) signifies that the energy estimates for the wave variables are coupled to L2 estimates for the derivatives of the eikonal function. This is a fundamental difficulty that one faces whenever one works with vectorfields constructed from an eikonal function adapted to the characteristics. We already stress here that a naive treatment of the term Y N trg/ χ in the energy estimates would result in the loss of a derivative that would preclude closure of the estimates. This is because crude estimates for Y N trg/ χ ∼ ∂ N +2 u based on the eikonal equation (1.7.1a) lead 41In

practice, we commute through µg since the µ-weighted wave operator exhibits better commutation ˘ Y }. properties with the elements of {L, X, 42In practice, one mostly relies on geometric decomposition formulas when decomposing the error terms in the commuted equations, rather than working with Cartesian components.

J. Speck

33

to an L2 estimate for ∂ N +2 u that depends on ∂ N +2 Ψ, which is one more derivative of Ψ (F ast) than is controlled by HN . However, the term Y N trg/ χ has a special tensorial structure and the loss of a derivative can be avoided through a procedure described below. As we explain below, we use this procedure only at the top-order43 because it leads to a rather degenerate top-order energy estimate, and we need improved estimates below top-order to close our proof. The improved estimates are possible because below top-order, one does not need to worry about the loss of a derivative. In fact, as we further explain below, to obtain the improved estimates, it is important that one should allow the loss of a derivative in the estimates for Y N trg/ χ below top-order. Taken together, these are unusual and technically challenging features of the study of shock formation that were first found in the works [1–4] of Alinhac44 and Christodoulou. We now provide some additional details on how to obtain a priori energy estimates. In the (F ast) (Slow) usual fashion, we must control RHSs (1.7.32a) and (1.7.32b) in terms of HM and HM (for suitable M ) so that we can use a version of Gronwall’s inequality. After a rather difficult Gronwall estimate, one obtains a hierarchy of energy estimates holding up the shock, which we now explain. The estimates feature the quantity

µ? (t, u) := min 1, min µ , u Σt

which essentially measures the worst case smallness for µ along Σut . As in all prior works on shock formation in more than one spatial dimension, our proof allows for the possibility that our high-order energies might blow up like negative powers of µ? , and, at the same time, guarantees that the energies become successively less singular as one reduces the number of derivatives. Moreover, one eventually reaches a level at which the energies remain uniformly bounded, all the way up to the shock; these non-degenerate energy estimates are what allows one to improve, via Sobolev embedding, the L∞ bootstrap assumptions that are crucial for all aspects of the proof. The hierarchy of energy estimates that we derive can be modeled

43More

precisely, in treating the most difficult top-order error terms involving Y N trg/ χ, we use this procedure only at the top-order. We also encounter less degenerate top-order error terms with factors that behave like µY N trg/ χ (see Lemma 9.11 and point (5) of Subsect. 5.2), and for these terms, thanks to the helpful factor of µ, it is permissible to use the procedure at all derivative levels. 44Actually, Alinhac’s approach allowed for some loss of differentiability stemming from his use of an eikonal function. As we mentioned in Subsect. 1.4, to overcome this difficulty, he used Nash–Moser estimates. In contrast, Christodoulou used an approach that avoided the derivative loss altogether, which is the approach we take in the present article.

Shock formation for quasilinear wave systems featuring multiple speeds 34 as follows: (F ast)

H18

(Slow)

(t, u), H18

(t, u) . ˚ 2 µ−11.8 (t, u), ?

(F ast) H17 (t, u),

(Slow) H17 (t, u)

(F ast) H13 (t, u),

(Slow) H13 (t, u)

(F ast)

H12

(Slow)

(t, u), H12

(F ast) H1 (t, u),

.˚ 2 µ−9.8 (t, u), ?

(1.7.33b)

··· ,

(1.7.33c)

(t, u), .˚ 2 µ−1.8 ?

(1.7.33d)

(t, u) . ˚ 2 ,

(Slow) H1 (t, u)

(1.7.33a)

(1.7.33e)

···

(1.7.33f)

.˚ 2 .

(1.7.33g)

The estimates (1.7.33a)-(1.7.33g) capture in spirit the energy estimates that we prove in this article; we refer the reader to Prop. 9.7 for the precise statements. The precise numerology behind the hierarchy (1.7.33a)-(1.7.33g) is intricate, but here are the main ideas: i) The toporder blowup-exponent of 11.8 found on RHS (1.7.33a) is tied to certain universal structural constants in the equations (such as the constant A appearing in (1.7.37) below) and is close to optimal by our approach; for example, if one considered data belonging to the Sobolev space H 100 , then our approach would only allow us to conclude that the energy controlling (t, u). ii) The fact that the estimates 100 derivatives of Ψ might blow up at the rate µ−11.8 ? becomes less singular by precisely two powers of µ? at each step in the descent seems to be fundamental. The root of this phenomenon is the following: an integration in time of 1−B (t, u); see (1.7.42). iii) µ−B ? (t, u) reduces the strength of the singularity by one degree to µ? To control error terms, it is convenient for the solutions to be such that slightly more than half of the energies are uniformly bounded up to the shock. Then, when we are bounding product error terms in L2 , we can exploit the fact that all but at-most-one factor in the product is uniformly bounded in L∞ up to the shock. We now discuss some of the main ideas behind deriving the energy estimate hierarchy (1.7.33a)-(1.7.33g). By far, the most difficult integrals to estimate are the ones on RHS (1.7.32a) involving Y N trg/ χ and some related ones that we have not displayed but that create similar difficulties. In the case of the scalar wave equations treated in [28], these difficult integrals were handled via extensions of techniques developed in [4]. In the present article, in treating these integrals, we encounter some new terms stemming from interactions between the fast wave, the eikonal function, and the slow wave; see the proof sketch of Prop. 8.2 for further discussion on this point. Later in this section, we will say a few words about these difficult integrals, but we will not discuss them in detail here since the most challenging aspects of these integrals were handled in [28]. Instead, in this subsubsection, ~ -involving integrals from RHSs (1.7.32a)we focus on describing the influence of the Y N W (1.7.32b) on the a priori estimates (1.7.33a)-(1.7.33g). Roughly, these integrals account for the self-interactions of the slow wave and the interaction of the slow wave with the fast wave up to the shock, which are the main new kinds of interactions accounted for in this paper; the remaining error integrals on RHS (1.7.32a) involve self-interactions of Ψ and the interaction of Ψ with the eikonal function, which were handled in [28], as well as interactions ~ with the below-top-order derivatives of the eikonal function, which are relatively easy of W to treat (see Lemma 9.13). Our main goal at present is the following:

J. Speck

35

~ -involving integrals from RHSs (1.7.32a)-(1.7.32b) We will sketch why the Y N W create only harmless exponential growth in the energies, which is allowable within our approach in view of our sufficiently good guess about the time of first shock formation (see the discussion following equation (1.7.6)) and the smallness of ˚ . ~ -involving integrals were the only types of error integrals that one In particular, if the Y N W encountered in the energy estimates, then at all derivatives levels, the energies would remain uniformly bounded by . ˚ 2 up to the shock. This shows that the interaction between the two waves is in some sense weak near the shock, even though the RHS of the slow wave equation (1.6.2b) contains ∂Ψ source terms that blow up at the shock. The weakness of the interaction is very much a “PDE effect” (it is not easily modeled by ODE inequalities) that is detectable only because our energies (1.7.30a)-(1.7.30b) contain non-µ-degenerate Put integrals and spacetime integrals. To proceed with our sketch, we let ¯ (F ast) (t, u) := H N

(F ast)

sup (t0 ,u0 )∈[0,t]×[0,u]

HN

(t0 , u0 ),

¯ (Slow) (t, u) := H N

sup (t0 ,u0 )∈[0,t]×[0,u]

(Slow)

HN

(t0 , u0 ).

We then note the following simple consequence of (1.7.30a)-(1.7.30b), (1.7.32a)-(1.7.32b), and Young’s inequality, where we are ignoring the Y N trg/ χ-involving integral on RHS (1.7.32a): Z u Z u (F ast) (F ast) 0 0 2 ¯ (Slow) (t, u0 ) du0 + · · · , (1.7.34) ¯ ¯ H H (t, u ) du + C H (t, u) ≤ C˚ +C N N N 0 0 u =0 u =0 Z u Z t (Slow) (F ast) 2 0 0 ¯ ¯ ¯ (Slow) (t0 , u) dt0 + · · · . HN (t, u) ≤ C˚ +C HN (t, u ) du + C H (1.7.35) N u0 =0

t0 =0

Then from (1.7.34)-(1.7.35) and Gronwall’s inequality in t and u, we conclude that ¯ (F ast) (t, u), H ¯ (Slow) (t, u) ≤ C˚ H 2 , N N

(1.7.36)

where, as we have mentioned, constants C are allowed to depend on ˚ δ−1 ∗ , the approximate time of first blowup (see (1.7.6) and the discussion below it). As we mentioned above, in reality, we are not able to prove the non-degenerate estimate (1.7.36) for large N because the Y N trg/ χ-involving integral on RHS (1.7.32a) leads to a much worse a priori energy estimate in the top-order case N = 18. This phenomenon is explained in detail in [28] for the case of a homogeneous scalar covariant wave equation g(Ψ) Ψ = 0; here we only describe the changes in the analysis of Y N trg/ χ compared to [28], the new feature being the presence of the inhomogeneous terms on RHS (1.6.2a). Let us first describe the estimate. Specifically, due to the difficult regularity theory of the eikonal function,45 we obtain (see below for more details) the following additional term on RHS (1.7.34) in the case N = 18: ! Z t Lµ (F ast) A (1.7.37) sup H18 (t0 , u) dt0 + · · · , u µ 0 Σ t =0

t0

where A is a universal positive constant that is independent of the structure of the nonlinearities and the number of times that the equations are commuted and 45Recall

that Y N trg/ χ ∼ ∂ N +2 u.

Shock formation for quasilinear wave systems featuring multiple speeds 36 · · · denotes similar or less degenerate error terms. By itself, the error term (1.7.37) would change the a priori estimate in a way that can roughly be described as follows: ¯ (F ast) (t, u), H ¯ (Slow) (t, u) ≤ C˚ H 2 µ−A 18 18 ? (t, u).

(1.7.38)

The factor of A on RHS (1.7.38) is partially responsible for the magnitude of the toporder blowup-exponent 11.8 on RHS (1.7.33a), though we stress that in a detailed proof, one encounters other types of degenerate error integrals that further enlarge the blowupexponents. Throughout the paper, we indicate the “important” structural constants that substantially contribute to the blowup-exponents by drawing boxes around them (see, for example, the RHS of the estimates of Prop. 9.6). The derivation of (1.7.38) as a consequence of the presence of the error term (1.7.37) is based on a difficult Gronwall estimate that requires having sharp information about the way that µ → 0 as well as the behavior of Lµ. Roughly, one can show (see Subsubsect. 1.7.5 for a discussion of the main ideas) that µ? (t, u) ∼ 1 − ˚ δ∗ t,

sup |Lµ| ∼ ˚ δ∗ ,

(1.7.39)

Σu t0

from which one may obtain (1.7.38) by Gronwall’s inequality (note that the same factor ˚ δ∗ defined in (1.7.6) appears in both expressions in (1.7.39)). We now explain the origin of the difficult error integral (1.7.37) and its connection to the following aforementioned difficulty: that of avoiding a loss of a derivative at the top-order when bounding the error term Y N trg/ χ in L2 . To proceed, we first note that using geometric decompositions,46 one obtains the following evolution equation for Y N trg/ χ, expressed here in schematic form (see the proof sketch of Prop. 8.2 for further discussion): LY N trg/ χ = LP N +1 Ψ + ∆ / P N Ψ + l.o.t.,

(1.7.40)

where ∆ / is the covariant Laplacian induced on `t,u by g(Ψ) and l.o.t. are lower-order terms ~ . In the top-order case N = 18, equation that do not involve the slow wave variable W (1.7.40) is not useful in its current form because the RHS involves one more derivative of Ψ than we can control by commuting equation (1.6.2a) 18 times and deriving energy estimates. To overcome this difficulty, we follow the following strategy, whose blueprint originates in the proof of the stability of Minkowski spacetime [5] and that was later used in the context of low-regularity well-posedness for wave equations [14] and finally in the context of shock formation [4]: one can decompose the fast wave equation (1.6.2a) using Prop. 2.12 and then algebraically replace µ∆ / P N Ψ (note the crucial factor of µ and see ˘ N Ψ + µLP N +1 Ψ + l.o.t. (written in schematic the first equality in (2.13.1b)) with LXP form) plus µ × RHS (1.6.2a). We can then bring these perfect L-derivative terms over to LHS (1.7.40) to obtain an evolution equation for a “modified” version of Y N trg/ χ, denoted by Modified, which can be written in the following schematic form: Modified }| { z N N N +1 ˘ L µY trg/ χ + XP Ψ + µP Ψ = µ × RHS (1.6.2a) + · · · ; (1.7.41) 46By

this, we essentially mean Raychaudhuri’s identity for the component RicLL of the Ricci curvature of the metric g(Ψ).

J. Speck

37

see (8.3.1) for the precise definition of the modified quantity. We stress that if we allowed g = g(Ψ, w) instead of g = g(Ψ), then our proof of (1.7.41) would break down in the sense that we would not generally be able to derive an analogous evolution equation featuring terms with allowable regularity on the RHS. To handle the first integral on RHS (1.7.32a), we now algebraically replace Y N trg/ χ = 1 1 ˘ N Modified − XP Ψ − P N +1 Ψ, which leads to three error integrals. After a bit of addiµ µ 1 ˘ N tional work, one finds that the integral corresponding to the second term − XP Ψ leads µ 1 to the integral in (1.7.37). The error integral corresponding to the first term Modified µ leads to a similar but more difficult error integral that we treat in inequality (9.8.4) and the discussion just below it. Note that in view of RHS (1.7.41), the L2 estimates for Modified are coupled to the inhomogeneous terms on the right-hand side of the fast wave equation ~ . That is, our reliance on a modified version (1.6.2a), which involve the slow wave variable W of trg/ χ leads to the coupling of the slow wave variable to the top-order estimates for the null mean curvature of the Pu . However, due to the presence of the factor of µ on RHS (1.7.41), the coupling terms are weak and are among the easier error terms to treat (see the proof sketch of Lemma 9.10). The error integral corresponding to the third term −P N +1 Ψ from the above algebraic decomposition is much easier to treat and is handled in Lemma 9.13. (F ast) from (1.7.33a) can blow up like We have now sketched why the top-order quantity H18 (Slow) 2 −11.8 ˚ as µZ? → 0. To understand why the same can occur for H18 , we simply consider µ? u ¯ (F ast) (t, u0 ) du0 on RHS (1.7.35) (in the case N = 18); the integration with the integral H u0 =0

18

(Slow)

respect to u0 does not ameliorate the strength of the singularity and thus H18 can blow (F ast) up at the same rate as H18 . It remains for us to explain why, in the energy hierarchy (1.7.33a)-(1.7.33g), the energy estimates become successively less singular with respect to powers of µ−1 ? at each stage in the descent. The main ideas are the same as in all prior works on shock formation in more than one spatial dimension. Specifically, at each level of derivatives, the strength of the singularity is driven by the Y N trg/ χ-involving integral on RHS (1.7.32a) and a few other integrals similar to it. The key point is that below top-order, we can estimate these integrals in a more direct fashion by integrating the RHS of the evolution equation (1.7.40) with respect to time (recall ∂ that L = ) to obtain an estimate for kY N trg/ χkL2 (Σut ) . Such an approach involves the loss of ∂t one derivative (which is permissible below top-order) and thus couples the below-top-order energy estimates to the top-order one. The gain is that the resulting error integrals are less singular with respect to powers of µ−1 ? compared to the top-order integral (1.7.37). We now provide a few more details in the just-below-top-order case N = 17 to illustrate the main ideas behind this “descent scheme.” The main idea is that the strength of the singularity is reduced with each integration in time due to the following estimates,47 valid for constants R1 estimates stated in (1.7.42) are a quasilinear version of the model estimates s=t s−B ds . t1−B R1 and s=t s−9/10 ds . 1, where B > 1 and 0 < t < 1 in the model estimates and t = 0 represents the time of first vanishing of µ? . 47The

Shock formation for quasilinear wave systems featuring multiple speeds 38 B > 1 (see Prop. 7.3 for the precise statements): Z

t

t0 =0

1 dt0 . µ?1−B (t0 , u), B µ? (t0 , u)

Z

t

1

9/10 0 (t , u) t0 =0 µ?

dt0 . 1,

(1.7.42)

which follow from having sharp information about the way in which µ? → 0 in time (see (1.7.39)). We now explain the role that the estimates (1.7.42) play in the descent scheme. Using the above strategy and (1.7.34), we obtain ¯ (F ast) (t, u) H 17

2

t

1

t0 =0

µ? (t0 , u)

Z

≤ C˚ +

1/2

Z q (F ast) 0 ¯ H (t , u)

t0

17

s=0

1 1/2

µ? (s, u)

q ¯ (F ast) (s, u) ds dt0 + · · · , H 18 (1.7.43)

q ¯ (F ast) (s, u) term corresponds to the loss of one where · · · denotes easier error terms, the H 18 1 derivative that one encounters in estimating kY 17 trg/ χkL2 (Σut0 ) , and the factors of 1/2 arise µ? from the µ-weights found in the energies (1.7.30a) along Σut hypersurfaces. In reality, when ¯ (F ast) , · · · as unknowns in a coupled ¯ (F ast) , H deriving a priori estimates, one must treat H 17 18 system of integral inequalities for which one derives a priori estimates via a complicated Gronwall argument. Here, instead of providing the lengthy technical details behind the Gronwall argument (which, as we describe in our proof sketch of Prop. 9.7, was already carried out in [28, Proposition 14.1]), to illustrate the main ideas, we demonstrate only the consistency of the integral inequality (1.7.43) with the less singular estimate (1.7.33b) (less singular compared to (1.7.33a), that is). Specifically, inserting the estimates (1.7.33a)(1.7.33b) into the double time integral on RHS (1.7.43) and using the first of (1.7.42) two times, we obtain ¯ (F ast) (t, u) H 17

2

2

Z

t

≤ C˚ +˚

≤ C˚ 2 + ˚ 2

t0 =0 Z t t0 =0

≤ C˚ 2 + ˚ 2

Z t0 1 1 ds dt0 + · · · 0 , u) 6.4 (s, u) µ5.4 (t µ s=0 ? ? 1 dt0 + · · · 10.8 µ? (t0 , u)

1 µ9.8 ? (t, u)

(1.7.44)

+ ··· .

Thus, the strength of the singularity on RHS (1.7.44) is at least consistent with the estimate (1.7.33b) that one aims to prove. Subsequent to obtaining the estimate (1.7.33b), one may continue the descent scheme, with the energies becoming successively less singular at each step in the descent. Eventually, one reaches a level (1.7.33e) at which, thanks to the second estimate in (1.7.42), one can show that the energies remain bounded all the way up to the shock. Finally, from the non-degenerate energy estimates (1.7.33e)-(1.7.33g) and Sobolev embedding (see Lemma 9.8), one can recover the non-degenerate L∞ estimates described in Subsubsect. 1.7.4, which, in a detailed proof, one needs to control various error terms and to show that µ vanishes in finite time (as we outlined in Subsubsect. 1.7.5).

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2. The Remaining Ingredients in the Geometric Setup When outlining our proof in Subsect. 1.7, we defined some basic geometric objects that we use in studying the solution. In this section, we construct most of the remaining such objects, exhibit their basic properties, and give rigorous definitions of most of the quantities that we informally referred to in Sect. 1. 2.1. Additional constructions related to the eikonal function. We recall that we constructed the eikonal function in Def. 1.1. To supplement the coordinates t and u, we now construct a local coordinate function on the tori `t,u (which are defined in Def. 1.3); the coordinate ϑ plays only a minor role in our analysis. Definition 2.1 (Geometric torus coordinate). We define the geometric torus coordinate ϑ to be the solution to the following initial value problem for a transport equation: (g −1 )αβ ∂α u∂β ϑ = 0,

(2.1.1)

2

(2.1.2)

ϑ|Σ0 = x .

Definition 2.2 (Geometric coordinates and partial derivatives). We refer to (t, u, ϑ) as the geometric coordinates, where t is the Cartesian time coordinate. We denote the corresponding geometric coordinate partial derivative vectorfields by ∂ ∂ ∂ , , Θ := . (2.1.3) ∂t ∂u ∂ϑ 2.2. Important vectorfields, the rescaled frame, and the unit frame. In this subsection, we construct some vectorfields that we use in our analysis and exhibit their basic properties. We start by defining the gradient vectorfield of the eikonal function: Lν(Geo) := −(g −1 )να ∂α u.

(2.2.1)

It is easy to see that L(Geo) is future-directed48 and g-null: g(L(Geo) , L(Geo) ) := gαβ Lα(Geo) Lβ(Geo) = 0.

(2.2.2)

Moreover, by differentiating equation (2.2.1) with DL(Geo) , where D is the Levi-Civita connection of g, one easily infers that L(Geo) is geodesic: DL(Geo) L(Geo) = 0

(2.2.3)

In addition, it is straightforward to see that L(Geo) is g-orthogonal to the characteristics Pu . Hence, the Pu have g-null normals, which justifies our use of the terminology null hypersurfaces in referring to them. It is convenient to work with a rescaled version of L(Geo) that we denote by L. It turns out that the Cartesian components Lα remain uniformly bounded up to the shock. Definition 2.3 (Rescaled null vectorfield). We define L := µL(Geo) . 48By

(2.2.4)

a future-directed vectorfield V , we mean that V 0 > 0, where V 0 is the “0” Cartesian component of V . Similarly, by a future-directed one-form ξ, we mean that the g-dual of ξ is future-directed. We analogously define past-directed vectofields and one-forms.

Shock formation for quasilinear wave systems featuring multiple speeds 40 Note that L is g-null since L(Geo) is. We also note that by (2.1.1), we have Lϑ = 0. ˘ and N , which are transversal to the characteristics We now define the vectorfields X, X, ˘ is rescaled by a factor of µ. Pu . For our subsequent analysis, it is important that X ˘ and N ). We define X to be the unique vectorfield that is Σt -tangent, Definition 2.4 (X, X, g-orthogonal to the `t,u , and normalized by g(L, X) = −1.

(2.2.5)

˘ := µX. X

(2.2.6)

N := L + X.

(2.2.7)

We define We define In our analysis, it is convenient to use the following two vectorfield frames. Definition 2.5 (Two frames). We define, respectively, the rescaled frame and the nonrescaled frame as follows: ˘ Θ}, {L, X, Rescaled frame, (2.2.8a) {L, X, Θ},

Non-rescaled frame.

(2.2.8b)

We now exhibit some basic properties of the above vectorfields. ˘ L, and N ] The following Lemma 2.1. [28, Lemma 2.1; Basic properties of X, X, identities hold: Lu = 0, ˘ = 1, Xu g(X, X) = 1, g(L, X) = −1,

Lt = L0 = 1, ˘ =X ˘ 0 = 0, Xt ˘ X) ˘ = µ2 , g(X, ˘ = −µ. g(L, X)

(2.2.9a) (2.2.9b) (2.2.10a) (2.2.10b)

Moreover, relative to the geometric coordinates, we have ∂ (2.2.11) L= . ∂t In addition, there exists an `t,u -tangent vectorfield Ξ = ξΘ (where ξ is a scalar function) such that ˘ = ∂ − Ξ = ∂ − ξΘ. X (2.2.12) ∂u ∂u The vectorfield N defined in (2.2.7) is future-directed, g-orthogonal to Σt and is normalized by g(N, N ) = −1.

(2.2.13)

Moreover, relative to Cartesian coordinates, we have (for ν = 0, 1, 2): N ν = −(g −1 )0ν .

(2.2.14)

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41

Finally, the following identities hold relative to the Cartesian coordinates (for ν = 0, 1, 2): X ν = −Lν − (g −1 )0ν ,

Xν = −Lν − δν0 ,

(2.2.15)

where δν0 is the standard Kronecker delta. 2.3. Projection tensorfields, G(F rame) , and projected Lie derivatives. Many of our constructions involve projections onto Σt and `t,u . Definition 2.6 (Projection tensorfields). We define the Σt -projection tensorfield Π and the `t,u -projection tensorfield Π / relative to Cartesian coordinates as follows: Πνµ := δνµ − Nν N µ = δνµ + δν0 Lµ + δν0 X µ , Π / νµ

µ

µ

µ

µ

µ

(2.3.1a) 0

µ

µ

:= δν + Xν L + Lν (L + X ) = δν − δν L + Lν X .

(2.3.1b)

Definition 2.7 (Projections of tensorfields). Given any spacetime tensorfield ξ, we define its Σt projection Πξ and its `t,u projection Π / ξ as follows: ···e µm ···µm (Πξ)µν11···ν := Πµeµ11 · · · Πµeµmm Πννe11 · · · Πννenn ξνeµe11···e νn , n

(2.3.2a)

···e µm ···µm (Π / ξ)µν11···ν := Π / µeµ11 · · · Π / ννe11 · · · Π / µeµmm Π / ννenn ξνeµe11···e νn . n

(2.3.2b)

We say that a spacetime tensorfield ξ is Σt -tangent (respectively `t,u -tangent) if Πξ = ξ (respectively if Π / ξ = ξ). Alternatively, we say that ξ is a Σt tensor (respectively `t,u tensor). Definition 2.8 (`t,u projection notation). If ξ is a spacetime tensor, then we define /ξ := Π / ξ. If ξ is a symmetric type define

0 2

(2.3.3)

spacetime tensor and V is a spacetime vectorfield, then we /ξ V := Π / (ξV ),

(2.3.4)

where ξV is the spacetime one-form with Cartesian components ξαν V α , (ν = 0, 1, 2). We often refer to the following arrays of `t,u -tangent tensorfields in our analysis. Definition 2.9 (Components of G and G0 relative to the non-rescaled frame). We define G(F rame) := (GLL , GLX , GXX , G /L , G /X , G /) to be the array of components of the tensorfield G defined in (1.6.8) relative to the nonrescaled frame (2.2.8b). Similarly, we define G0(F rame) to be the analogous array for the tensorfield G0(F rame) defined in (1.6.8). Throughout, LV ξ denotes the Lie derivative of the tensorfield ξ with respect to V . If V and W are both vectorfields, then we often use the standard Lie bracket notation [V, W ] := LV W . In our analysis, we often differentiate various quantities with the projected Lie derivatives from the following definition. Definition 2.10 (`t,u and Σt -projected Lie derivatives). Given a tensorfield ξ and a vectorfield V , we define the Σt -projected Lie derivative LV ξ of ξ and the `t,u -projected Lie derivative L /V ξ of ξ as follows: LV ξ := ΠLV ξ,

L /V ξ := Π / LV ξ.

(2.3.5)

Shock formation for quasilinear wave systems featuring multiple speeds 42 2.4. First and second fundamental forms, the trace of a tensorfield, covariant differential operators, and the geometric torus differential. Definition 2.11 (First fundamental forms). We define the first fundamental form g of Σt and the first fundamental form g/ of `t,u as follows: g := Πg,

g/ := Π / g.

(2.4.1)

We define the corresponding inverse first fundamental forms by raising the indices with g −1 : (g −1 )µν := (g −1 )µα (g −1 )νβ g αβ ,

(g/−1 )µν := (g −1 )µα (g −1 )νβ g/αβ . (2.4.2) Definition 2.12 (g/-trace of a tensorfield). If ξ is a type 02 `t,u −tangent tensor, then trg/ ξ := (g/−1 )αβ ξαβ denotes its g/−trace. Definition 2.13 (Differential operators associated to the metrics). We use the following notation for various differential operators associated to the spacetime metric g and the Riemannian metric g/ induced on the `t,u . • D denotes the Levi-Civita connection of the spacetime metric g. • ∇ / denotes the Levi-Civita connection of g/. • If ξ is an `t,u -tangent one-form, then div / ξ is the scalar-valued function div / ξ := g/−1 ·∇ / ξ. −1 • Similarly, if V is an `t,u -tangent vectorfield, then div / V := g/ · ∇ / V[ , where V[ is the one-form g/-dual to V . • If ξ is a symmetric type 02 `t,u -tangent tensorfield, then div / ξ is the `t,u -tangent oneform div / ξ := g/−1 · ∇ / ξ, where the two contraction indices in ∇ / ξ correspond to the operator ∇ / and the first index of ξ. Definition 2.14 (Covariant wave operator and Laplacian). We use the following standard notation. 2 • g := (g −1 )αβ Dαβ denotes the covariant wave operator corresponding to the spacetime metric g. • ∆ / := g/−1 · ∇ / 2 denotes the covariant Laplacian corresponding to g/. Definition 2.15 (Geometric torus differential). If f is a scalar function on `t,u , then d/f := ∇ /f = Π / Df , where Df is the gradient one-form associated to f . Def. 2.15 allows us to avoid potentially confusing notation such as ∇ / Li by instead using d/Li ; the latter notation clarifies that Li is to be viewed as a scalar Cartesian component function under differentiation. Definition 2.16 (Second fundamental forms). We define the second fundamental form 0 k of Σt , which is a symmetric type 2 Σt -tangent tensorfield, by 1 k := LN g. (2.4.3) 2 We define the null second fundamental form χ of `t,u , which is a symmetric type 02 `t,u -tangent tensorfield, by 1 χ := L / g/. (2.4.4) 2 L

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43

2.5. Identities for various tensorfields. We now provide some identities for various `t,u tensorfields. Lemma 2.2. [28, Lemma 2.3; Alternate expressions for the second fundamental forms] We have the following identities: χΘΘ = g(DΘ L, Θ),

/kXΘ = g(DΘ L, X).

(2.5.1a)

In the next lemma, we decompose some `t,u -tangent tensorfields that arise in our analysis. Lemma 2.3. [28, Lemma 2.13; Expressions for ζ and /k ] Let ζ be the `t,u one-form defined by ζΘ := /kXΘ .

(2.5.2)

We have the following identities for the `t,u tensorfields /k and ζ: ζ = µ−1 ζ(T rans−Ψ) + ζ(T an−Ψ) ,

(2.5.3a)

/k = µ−1/k (T rans−Ψ) + /k (T an−Ψ) ,

(2.5.3b)

where 1 ˘ ζ(T rans−Ψ) := − G / XΨ, 2 L 1 ˘ / XΨ, /k (T rans−Ψ) := G 2

(2.5.4a) (2.5.4b)

and 1 1 1 ζ(T an−Ψ) := G / X LΨ − GLX d/Ψ − GXX d/Ψ, 2 2 2 1 1 1 1 1 (T an−Ψ) /k := G / LΨ − G / L ⊗ d/Ψ − d/Ψ ⊗ G /L − G / X ⊗ d/Ψ − d/Ψ ⊗ G /X . 2 2 2 2 2 2.6. Metric decompositions.

(2.5.5a) (2.5.5b)

Lemma 2.4. [28, Lemma 2.4; Expressions for g and g −1 in terms of the non-rescaled frame] We have the following identities: gµν = −Lµ Lν − (Lµ Xν + Xµ Lν ) + g/µν ,

(2.6.1a)

(g −1 )µν = −Lµ Lν − (Lµ X ν + X µ Lν ) + (g/−1 )µν .

(2.6.1b)

The following scalar function captures the `t,u part of g. Definition 2.17 (The metric component υ). We define the scalar function υ > 0 by υ 2 := g(Θ, Θ) = g/(Θ, Θ).

(2.6.2)

2.7. The change of variables map. In this subsection, we define the change of variables map between the geometric and Cartesian coordinates and illustrate some of its basic properties. Definition 2.18. We define Υ : [0, T ) × [0, U0 ] × T → MT,U0 , Υ(t, u, ϑ) := (t, x1 , x2 ), to be the change of variables map from geometric to Cartesian coordinates.

Shock formation for quasilinear wave systems featuring multiple speeds 44 Lemma 2.5. [28, Lemma 2.7; Basic properties of the change of variables map] We have the following expression for the Jacobian of Υ: 1 0 0 0 1 2 ∂(x , x , x ) 1 ˘ 1 ∂Υ L X + Ξ1 Θ1 . := = (2.7.1) ∂(t, u, ϑ) ∂(t, u, ϑ) 2 2 2 2 ˘ L X +Ξ Θ Moreover, we have det

∂(x1 , x2 ) ∂(x0 , x1 , x2 ) = det = µ(detg ij )−1/2 υ, ∂(t, u, ϑ) ∂(u, ϑ)

(2.7.2)

where υ is the metric component from Def. 2.17 and (detg ij )−1/2 is a smooth function of Ψ in a neighborhood of 0 with (detg ij )−1/2 (Ψ = 0) = 1. In (2.7.2), g is viewed as the Riemannian metric on ΣUt 0 defined by (2.4.1) and detg ij is the determinant of the corresponding 2 × 2 matrix of components of g relative to the Cartesian spatial coordinates. 2.8. Commutation vectorfields and a basic vectorfield commutation identity. In this subsection, we define the commutation vectorfields that we use when commuting equations to obtain estimates for the solution’s derivatives. Definition 2.19 (The vectorfields Y(F lat) and Y ). We define the Cartesian components of the Σt -tangent vectorfields Y(F lat) and Y as follows (i = 1, 2): i i Y(F lat) := δ2 , i

Y :=

a Π / ai Y(F lat)

(2.8.1) =

Π / 2i ,

(2.8.2)

where Π / is the `t,u projection tensorfield defined in (2.3.1b). To derive estimates for the solution’s derivatives, we commute the equations with the elements of the following set of vectorfields. Definition 2.20 (Commutation vectorfields). We define the commutation set Z as follows: ˘ Y }, Z := {L, X,

(2.8.3)

˘ and Y are respectively defined by (2.2.4), (2.2.6), and (2.8.2). where L, X, We define the Pu -tangent commutation set P as follows: P := {L, Y }.

(2.8.4)

We use the following commutation identity throughout our analysis. ˘ Y commute with d/] For scalar functions f and Lemma 2.6. [28, Lemma 2.10; L, X, ˘ Y }, we have V ∈ {L, X, L /V d/f = d/V f. The following quantities are convenient to study because they are small.

(2.8.5)

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45

Definition 2.21 (Perturbed part of various vectorfields). For i = 1, 2, we define the following scalar functions: Li(Small) := Li − δ1i ,

i X(Small) := X i + δ1i ,

i Y(Small) := Y i − δ2i .

(2.8.6)

The vectorfields L, X, and Y in (2.8.6) are defined in Defs. 2.3, 2.4, and 2.19. In the next lemma, we characterize the discrepancy between Y(F lat) and Y . Lemma 2.7. [28, Lemma 2.8; Decomposition of Y(F lat) ] We can decompose Y(F lat) into an `t,u -tangent vectorfield and a vectorfield parallel to X as follows: since Y is `t,u -tangent, there exists a scalar function ρ such that i i i Y(F lat) = Y + ρX , i Y(Small)

(2.8.7a)

i

= −ρX .

(2.8.7b)

Moreover, we have (Small)

a b a ρ = g(Y(F lat) , X) = gab Y(F lat) X = g2a X = g21

2 X 1 − g22 X(Small) .

(2.8.8)

2.9. Deformation tensors. In this subsection, we provide the standard definition of the deformation tensor of a vectorfield. Definition 2.22 (Deformation tensor of a vectorfield V ). If V is a spacetime vectorfield, (V ) then its deformation tensor π (relative to the spacetime metric g) is the symmetric type 0 spacetime tensorfield 2 παβ := LV gαβ = Dα Vβ + Dβ Vα ,

(V )

(2.9.1)

where the last equality in (2.9.1) is a well-known consequence of the torsion-free property of the connection D. 2.10. Transport equations for the eikonal function quantities. In this subsection, we provide the main evolution equations that we use to control µ, the Cartesian component functions Li(Small) , χ, and their derivatives, except at the top order. We sometimes refer to these tensorfields as the eikonal function quantities since they are constructed out of the eikonal function. To control their top-order derivatives, one needs to rely on the modified quantities described in Subsubsect. 1.7.6. Lemma 2.8. [28, Lemma 2.12; The transport equations verified by µ and Li ] The inverse foliation density µ defined in (1.7.2) verifies the following transport equation: 1 ˘ − 1 µGLL LΨ − µGLX LΨ. Lµ = GLL XΨ (2.10.1) 2 2 Moreover, the scalar-valued Cartesian component functions Li(Small) , (i = 1, 2), defined in (2.8.6), verify the following transport equation: 1 1 1 LLi(Small) = − GLL (LΨ)Li − GLL (LΨ)(g −1 )0i − G /L# · (d/xi )(LΨ) + GLL (d/# Ψ) · d/xi . 2 2 2 (2.10.2) ˘ i The next lemma provides a useful identity for XL (Small) .

Shock formation for quasilinear wave systems featuring multiple speeds 46 ˘ i ] We have the following identity for the Lemma 2.9. [28, Lemma 2.10; Formula for XL scalar-valued functions Li(Small) , (i = 1, 2): 1 1 1 ˘ − GLL XΨ + µGLL LΨ + µGLX LΨ + µGXX LΨ Li (2.10.3) 2 2 2 1 1 1 ˘ + µGLL LΨ + µGLX LΨ + µGXX LΨ (g −1 )0i + − GLL XΨ 2 2 2 ˘ + 1 µGXX d/# Ψ · d/xi + (d/# µ) · d/xi . − G /L# XΨ 2

˘ i XL (Small) =

2.11. Useful expressions for the null second fundamental form. The identities provided by the following lemma are convenient for deriving estimates for χ and related quantities. Lemma 2.10. [28, Lemma 2.15; Identities involving χ] We have the following identities: 1 χ = gab (d/La ) ⊗ d/xb + G / LΨ, 2 1 trg/ χ = gab g/−1 · (d/La ) ⊗ d/xb + g/−1 · G / LΨ, 2 L ln υ = trg/ χ,

(2.11.1a) (2.11.1b) (2.11.1c)

where χ is the `t,u -tangent tensorfield defined by (2.4.4) and υ is the metric component from Def. 2.17. 2.12. Arrays of unknowns and schematic notation. We use the following arrays for convenient shorthand notation. Definition 2.23 (Shorthand notation for the fast wave and the eikonal function quantities). We define the following arrays γ and γ of scalar functions: γ := Ψ, L1(Small) , L2(Small) ,

(2.12.1a)

γ := Ψ, µ − 1, L1(Small) , L2(Small) .

(2.12.1b)

Remark 2.1 (Schematic functional dependence). In the remainder of the article, we use the notation f(ξ(1) , ξ(2) , · · · , ξ(m) ) to schematically depict an expression (often tensorial and involving contractions) that depends smoothly on the `t,u -tangent tensorfields ξ(1) , ξ(2) , · · · , ξ(m) . Note that in general, f(0) 6= 0. Remark 2.2 (The meaning of the symbol P ). Throughout, P schematically denotes a differential operator that is tangent to the characteristics Pu , such as L, Y , or d/. For example, P f might denote d/f or Lf . We use such notation when the precise details of P are not important.

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47

Lemma 2.11 (Schematic structure of various tensorfields). We have the following schematic relations for scalar functions: gαβ , (g −1 )αβ , (g −1 )αβ , g/αβ , (g/−1 )αβ , Gαβ , G0αβ , Π / βα , Lα , X α , Y α = f(γ), GLL , GLX , GXX , G0LL , G0LX , G0XX (Small)

gαβ

= f(γ),

α α , Y(Small) , X(Small) , ρ = f(γ)γ, ˘ α = f(γ). X

(2.12.2a) (2.12.2b) (2.12.2c) (2.12.2d)

Moreover, we have the following schematic relations for `t,u -tangent tensorfields: g/, G /L , G /X , G /,G / L0 , G / X0 , G / 0 = f(γ, d/x1 , d/x2 ), −1

1

(2.12.3a) 2

Y = f(γ, g/ , d/x , d/x ),

(2.12.3b)

ζ(T an−Ψ) , /k (T an−Ψ) = f(γ, d/x1 , d/x2 )P Ψ, ˘ ζ(T rans−Ψ) , /k (T rans−Ψ) = f(γ, d/x1 , d/x2 )XΨ,

(2.12.3c) (2.12.3d)

χ = f(γ, d/x1 , d/x2 )P γ,

(2.12.3e)

−1

(2.12.3f)

1

2

trg/ χ = f(γ, g/ , d/x , d/x )P γ.

Finally, the null form Qg (∂Ψ, ∂Ψ) defined in (1.6.3), upon being multiplied by µ, has the following schematic structure: ˘ P Ψ)P Ψ. µQg (∂Ψ, ∂Ψ) = f(γ, XΨ,

(2.12.4)

Proof. All relations were proved in [28, Lemma 2.19] except for (2.12.4), which was proved in [16, Lemma 3.19]. 2.13. Frame decomposition of the wave operator. In the following proposition, we decompose µg(Ψ) f relative to the rescaled frame (2.2.8a). Proposition 2.12. [28, Proposition 2.16; Frame decomposition of µg(Ψ) f ] Let f be a ˘ Θ}, µg(Ψ) f can be expressed in scalar function. Then relative to the rescaled frame {L, X, either of the following two forms: ˘ ) + µ∆ ˘ − µtrg//k Lf − 2µζ# · d/f, µg(Ψ) f = −L(µLf + 2Xf / f − trg/ χXf (2.13.1a) # # ˘ ˘ − ωLf + 2µζ · d/f + 2(d/ µ) · d/f, = −(µL + 2X)(Lf ) + µ∆ / f − trg/ χXf (2.13.1b) where the `t,u -tangent tensorfields χ, ζ, and /k can be expressed via (2.11.1a), (2.5.3a), and (2.5.3b). 2.14. Relationship between Cartesian and geometric partial derivative vectorfields. In the next lemma, we provide explicit expressions for the Cartesian coordinate partial derivative vectorfields ∂ν as linear combinations of the commutation vectorfields of Def. 2.20. Lemma 2.13. [16, Lemma 3.17; Expression for ∂ν in terms of geometric vectorfields] We can express the Cartesian coordinate partial derivative vectorfields in terms of L, X, and

Shock formation for quasilinear wave systems featuring multiple speeds 48 Y as follows, (i = 1, 2): ga0 Y a ∂t = L − (gα0 L )X + gcd Y c Y d gai Y a a ∂i = (gai X )X + Y. gcd Y c Y d

α

Y,

(2.14.1a) (2.14.1b)

2.15. An algebraic expression for the transversal derivative of the slow wave. We ˘W ~ in terms of Pu -tangential will use the following algebraic lemma in order to control X ~ and other simple error terms. derivatives of W ˘W ~ ). Equations (1.6.11a)-(1.6.11c) imply the Lemma 2.14 (Algebraic expression for X following schematic algebraic relation, where the f depend smoothly on their arguments when~ | is sufficiently small: ever |γ| + |W ~ XΨ, ˘ P Ψ)P Ψ + f(γ, W, ~ XΨ, ˘ P Ψ)P W ~ ˘W ~ = f(γ, W, X ~ XΨ, ˘ P Ψ)Ψ + f(γ, W, ~ XΨ, ˘ P Ψ)W. ~ + f(γ, W,

(2.15.1)

~ = Proof. We first write the sub-system (1.6.11a)-(1.6.11c) in the matrix-vector form µAα ∂α W ~ ), A0 is the 4 × 4 identity matrix, and F corresponds to the inµF , where Aα = Aα (Ψ, W homogeneous terms on the second line of RHS (1.6.11a). It is straightforward to check (say, at an arbitrary given point p, relative to a frame in which h|p = diag(−1, 1, 1)) ~ = that if ξ is h-timelike,49 then the matrix Aα ξα is invertible. Decomposing µAα ∂α W ~ + α2 X ˘W ~ + µα3 Y W ~ , where the αi are 4 × 4 matrices, we compute, with the help µα1 LW of Lemma 2.13 and (2.2.15), that α2 = −A0 L0 + Aa Xa = −Aα Lα . Since L is g-null, we deduce from (1.6.13) that the one-form Lα is h-timelike. Thus, from the above observations, we see that α2 is invertible. Moreover, with the help of Lemma 2.11, we deduce that ~ ), (i = 1, 2, 3). The desired relation (2.15.1) is a simple consequence of these αi = f(γ, W facts, the assumptions on the inhomogeneous terms stated in (1.6.5), and Lemma 2.11. 2.16. Geometric integration. We define our geometric integrals in terms of length, area, and volume forms that remain non-degenerate throughout the evolution, all the way up to the shock. Definition 2.24 (Non-degenerate forms and related integrals). We define the length form dλg/ on `t,u , the area form d$ on Σut , the area form d$ on Put , and the volume form d$ on Mt,u as follows (relative to the geometric coordinates): d$ = d$(t, u0 , ϑ) := dλg/ (t, u0 , ϑ)du0 ,

dλg/ = dλg/ (t, u, ϑ) := υ(t, u, ϑ)dϑ, 0

0

0

d$ = d$(t , u, ϑ) := dλg/ (t , u, ϑ)dt ,

h-timelike, we mean that (h−1 )αβ ξα ξβ < 0.

0

0

0

0

(2.16.1) 0

d$ = d$(t , u , ϑ) := dλg/ (t , u , ϑ)du dt ,

where υ is the scalar function from Def. 2.17. 49By

0

J. Speck

49

If f is a scalar function, then we define Z Z f dλg/ := f (t, u, ϑ) υ(t, u, ϑ)dϑ, `t,u ϑ∈T Z Z u Z f d$ := f (t, u0 , ϑ) υ(t, u0 , ϑ)dϑdu0 , Σu t

u0 =0 ϑ∈T Z t Z

Z f d$ :=

t0 =0

Put

Z

Z

t

ϑ∈T Z u

t0 =0

u0 =0

(2.16.2b)

f (t0 , u, ϑ) υ(t0 , u, ϑ)dϑdt0 , Z

f d$ := Mt,u

(2.16.2a)

(2.16.2c)

f (t0 , u0 , ϑ) υ(t0 , u0 , ϑ)dϑdu0 dt0 .

(2.16.2d)

ϑ∈T

Remark 2.3. One can check that the canonical forms associated to g and g are, respectively, µd$ and µd$. 2.17. Integration with respect to Cartesian forms. In deriving energy identities for the ~ , it is convenient to carry out calculations relative to the Cartesian slow wave variables W coordinates. In this subsection, we define some basic objects that play a role in these identities. Definition 2.25 (The one-form H). We define H to be the one-form with the following Cartesian components: 1 Hα := − κλ Lα , (2.17.1) (δ Lκ Lλ )1/2 where δ κλ is the standard inverse Euclidean metric on R × Σ (that is, δ κλ = diag (1,1,1) relative to the Cartesian coordinates). Note that H is the Euclidean-unit-length co-normal to Pu . Definition 2.26 (Euclidean volume and area forms and related integrals). We define dM := dx1 dx2 dt,

dΣ := dx1 dx2 ,

dP

to be, respectively, the standard volume form on Mt,u induced by the Euclidean metric50 on R × Σ, the standard area form induced on Σt by the Euclidean metric on R × Σ, and the standard area form induced on Pu by the Euclidean metric on R × Σ. Remark 2.4 (We do not use the Euclidean forms when deriving estimates). We could of course provide explicit expressions for the integrals of functions over various Euclidean forms of Def. 2.26. For example, we have R domainsR with respect to the 1 2 f dΣ = f (t, x , x ) dx1 dx2 . We avoid providing further detailed expresΣU {0≤u(t,x1 ,x2 )≤U } t sions because we do not need them. The reason is that we never estimate integrals involving the Euclidean volume forms; before deriving estimates, we will always use Lemma 2.15 below order to replace the Euclidean forms with the geometric ones of Def. 2.24; we use the Euclidean forms only when deriving energy identities relative to the Cartesian coordinates, in which the Euclidean forms naturally appear. 50By

definition, the Euclidean metric has the components diag(1, 1, 1) relative to the standard Cartesian coordinates (t, x1 , x2 ) on R × Σ.

Shock formation for quasilinear wave systems featuring multiple speeds 50 2.18. Comparison between the Cartesian forms and the geometric forms. After we derive energy identities for the slow wave relative to the Cartesian coordinates, it will be convenient for us to express the corresponding integrals in terms of the geometric forms. The following lemma provides some identities that are useful in this regard. Lemma 2.15 (Comparison between Euclidean and geometric forms). There exist scalar functions, schematically denoted by f(γ), that are smooth for |γ| sufficiently small and such that the following relationship holds between the geometric forms of Def. 2.24 and the Euclidean forms of Def. 2.26: n√ o dM = µ {1 + γf(γ)} $, dΣ = µ {1 + γf(γ)} $, dP = 2 + γf(γ) $. (2.18.1) √ Proof. We prove only the identity dP = 2 + γf(γ) $ since the other two identities in (2.18.1) are a straightforward consequence of Lemma 2.5 (in particular, the Jacobian determinant expressions in (2.7.2)). Throughout this proof, as a slight abuse of notation, we view $ (see (2.16.1)) to be the two-form υdt ∧ dϑ on Pu , where dt ∧ dϑ = dt ⊗ dϑ − dϑ ⊗ dt. Similarly, we view dP to be the two-form induced on Pu by the standard Euclidean metric on R × Σ. Then relative to Cartesian coordinates, we have dP = (dx0 ∧ dx1 ∧ dx2 ) · V , where V is the future-directed Euclidean normal to Pu and (dx0 ∧ dx1 ∧ dx2 ) · V denotes contraction of V against the first slot of dx0 ∧ dx1 ∧ dx2 . Note that V α = δ αβ Hβ , where Hα is defined in (2.17.1) and δ αβ = diag (1,1,1) is the standard inverse Euclidean metric on R × Σ. Since $ and dP are proportional and since dt ∧ dϑ · (L ⊗ Θ) = 1, it suffices to show that √ 2 + γf(γ) υ = (dx0 ∧ dx1 ∧ dx2 ) · (V ⊗ L ⊗ Θ). To proceed, we note (dx0 ∧ dx1 ∧ dx2 ) · that V 0 L0 0 (V ⊗ L ⊗ Θ) is equal to the determinant of the 3 × 3 matrix N := V 1 L1 Θ1 . Next, V 2 L2 Θ2 we consider the 3 × 3 matrix M := N > · g · N , where we view g as a 3 × 3 matrix expressed relative to the Cartesian coordinates. Since (1.6.6)-(1.6.7) imply that |detg| = 1 + γf(γ) relative to the Cartesian coordinates and since |detM | = (detN )2 , the desired conclusion will follow once we prove |detM | = υ 2 {2 + γf(γ)}. To obtain this relation, we first compute g(V, V ) g(V, L) g(V, Θ) and thus detM = − (g(L, V ))2 υ 2 . Finally, from 0 0 that M = g(L, V ) g(Θ, V ) 0 υ2 (1.6.6), (1.6.7), (2.8.6), and (2.17.1), we compute (relative to the Cartesian coordinates) √ that g(L, V ) = − 2 + γf(γ), which, in conjunction with the above computations, yields |detM | = υ 2 {2 + γf(γ)} as desired. 3. Norms, Initial Data, Bootstrap Assumptions, and Smallness Assumptions In this section, we state our size assumptions on the data, formulate appropriate bootstrap assumptions for the solution, and state our smallness assumptions. As we mentioned in the introduction, the solutions under study here correspond to perturbations of simple outgoing plane waves. In [28, Remark 7.6], in the case of scalar quasilinear wave equations g(Ψ) Ψ = 0, the authors sketched a proof that there exist initial data that are compactly supported in Σ10 and that verify the desired assumptions. The arguments given there are straightforward

J. Speck

51

but tedious. Thus, in the present article, we note only that similar arguments can be used to prove the existence of data for the system (1.6.2a) + (1.6.11a)-(1.6.11d) that verify the size assumptions of the present article. Actually, in the present article, the solutions are allowed to be non-zero along P0 , which is different than in [28]. However, this difference is minor. We also note that a proof of the existence of data similar to the data treated here (in particular, the existence of data that are allowed to be non-zero along P0 ) was sketched in [16] in the context of the compressible Euler equations with vorticity. 3.1. Norms. We will derive estimates for scalar functions and `t,u -tangent tensorfields. We use the metric g/ when taking the pointwise norm of `t,u -tangent tensorfields, a concept that we make precise in the next definition. ···µm Definition 3.1 (Pointwise norms). If ξνµ11···ν is a type m `t,u tensor, then we define the n n norm |ξ| ≥ 0 by ···e µm ···µm µ |ξ|2 := g/µ1 µe1 · · · g/µm µem (g/−1 )ν1 νe1 · · · (g/−1 )νn νen ξνµ11···ν ξνee11···e νn . n

(3.1.1)

We use L2 and L∞ norms in our analysis. Definition 3.2 (L2 and L∞ norms). In terms of the non-degenerate forms of Def. 2.24, we define the following norms for `t,u -tangent tensorfields: Z Z 2 2 2 kξkL2 (`t,u ) := |ξ| dλg/ , kξkL2 (Σut ) := |ξ|2 d$, (3.1.2a) u `t,u Σt Z 2 kξkL2 (Put ) := |ξ|2 d$, Put

kξkL∞ (`t,u ) := ess supϑ∈T |ξ|(t, u, ϑ),

kξkL∞ (Σut ) := ess sup(u0 ,ϑ)∈[0,u]×T |ξ|(t, u0 , ϑ), (3.1.2b)

kξkL∞ (Put ) := ess sup(t0 ,ϑ)∈[0,t]×T |ξ|(t0 , u, ϑ). Remark 3.1 (Subset norms). We sometimes use norms k · kL2 (Ω) and k · kL∞ (Ω) , where Ω is a subset of Σut . These norms are defined by replacing Σut with Ω in (3.1.2a) and (3.1.2b). 3.2. Strings of commutation vectorfields and vectorfield seminorms. We use the following shorthand notation to capture the relevant structure of our vectorfield operators and to schematically depict estimates. Remark 3.2. Some operators in Def. 3.3 are decorated with a ∗. These operators involve Pu -tangent differentiations that often lead to a gain in smallness in the estimates. More precisely, the operators P∗N always lead to a gain in smallness while the operators Z∗N ;M lead to a gain in smallness except perhaps when they are applied to µ (because Lµ and its ˘ derivatives are not generally small). X Definition 3.3 (Strings of commutation vectorfields and vectorfield seminorms). • Z N ;M f denotes an arbitrary string of N commutation vectorfields in Z (see (2.8.3)) applied to f , where the string contains at most M factors of the Put -transversal ˘ vectorfield X.

Shock formation for quasilinear wave systems featuring multiple speeds 52 • P N f denotes an arbitrary string of N commutation vectorfields in P (see (2.8.4)) applied to f . • For N ≥ 1, Z∗N ;M f denotes an arbitrary string of N commutation vectorfields in Z applied to f , where the string contains at least one Put -tangent factor and at most ˘ We also set Z∗0;0 f := f . M factors of X. • For N ≥ 1, P∗N f denotes an arbitrary string of N commutation vectorfields in P applied to f , where the string contains at least one factor of Y or at least two factors of L. • For `t,u -tangent tensorfields ξ, we similarly define strings of `t,u -projected Lie deriva;M tives such as L /N ξ. Z We also define pointwise seminorms constructed out of sums of the above strings of vectorfields: • |Z N ;M f | simply denotes the magnitude of one of the Z N ;M f as defined above (there is no summation). 0 0 • |Z ≤N ;M f | is the sum over all terms of the form |Z N ;M f | with N 0 ≤ N and Z N ;M f as defined above. When N = M = 1, we sometimes write |Z ≤1 f | instead of |Z ≤1;1 f |. 0 • |Z [1,N ];M f | is the sum over all terms of the form |Z N ;M f | with 1 ≤ N 0 ≤ N and 0 Z N ;M f as defined above. [1,N ] ≤N ;M ˘ [1,N ] f |, etc., are defined analogously. For • Sums such as |P∗ f |, |L /Z ξ|, |Y ≤1 f |, |X N copies z }| { ˘ [1,N ] f | = |Xf ˘ | + |X ˘ Xf ˘ |+···+|X ˘X ˘ ···X ˘ f |. We write |P∗ f | instead of example, |X [1,1] |P∗ f |.

3.3. Assumptions on the initial data and the behavior of quantities along Σ0 . As we mentioned in Subsubsect. 1.7.2, the reciprocal of the parameter ˚ δ∗ > 0 is approximately equal to the time of first shock formation.

Definition 3.4 (The quantity that controls the blowup-time). We define

h i 1 ˚ ˘ δ∗ := sup GLL XΨ . 2 Σ10 −

We assume that the data verify the following size assumptions.

(3.3.1)

J. Speck

53

Assumptions along Σ10 .

≤19;≤1

Z∗ Ψ L2 (Σ1 ) 0

≤18 ~

P W 2 1 L (Σ0 )

≤10;≤2

≤11;≤1

˘ ˘ ˘

Z∗ Ψ L∞ (Σ1 ) , LX X XΨ Ψ L∞ (Σ1 ) , Z∗ 0 0 L∞ (Σ10 )

≤10 ~

~

P W ∞ 1 , Z ≤9;≤1 W

∞ 1 L (Σ0 ) L (Σ0 )

˘ [1,3]

X Ψ ∞ 1 L (Σ0 )

˘ ˘~

X X W ∞ 1

≤˚ ,

(3.3.2)

≤˚ ,

(3.3.3)

≤˚ ,

(3.3.4)

≤˚ ,

(3.3.5)

≤˚ δ,

(3.3.6)

≤˚ δ.

(3.3.7)

L (Σ0 )

˚−1

Assumptions along P02δ∗ .

≤19

P Ψ

, ≤˚ ,

(3.3.8)

≤18 ~

P W

≤˚ ,

(3.3.9)

≤17

P Ψ

≤˚ ,

(3.3.10)

≤16 ~

P W

≤˚ .

(3.3.11)

2˚ δ−1 L2 P0 ∗

2˚ δ−1 L2 P0 ∗

2˚ δ−1 L∞ P0 ∗

2˚ δ−1 L∞ P0 ∗

Assumptions along `t,0 . We assume that for t ∈ [0, 2˚ δ−1 ∗ ], we have

≤1

Z Ψ ∞ ≤˚ . L (`t,0 ) Assumptions along `0,u . We assume that for u ∈ [0, 1], we have

≤18

P Ψ 2 ≤˚ L (` )

0,u

≤17 ~ ≤˚ .

P W 2

(3.3.12)

(3.3.13) (3.3.14)

L (`0,u )

3.4. Initial behavior of the eikonal function quantities. The data-size assumptions of Subsect. 3.3 determine the initial size of various quantities constructed out of the eikonal function. In the next lemma, we estimate these data-dependent quantities in various norms. ˚−1

Lemma 3.1 (Behavior of the eikonal function quantities along Σ10 and P02δ∗ ). For initial data verifying the assumptions of Subsect. 3.3, the following L2 and L∞ estimates ˚−1 hold along Σ10 and P02δ∗ whenever ˚ is sufficiently small, where the implicit constants are ˚ allowed to depend on δ (see Subsect. 3.2 regarding the vectorfield operator notation):

≤19;3 i

˘ [1,3] i

Z∗ L(Small) L2 (Σ1 ) . ˚ , X L (3.4.1)

(Small) 2 1 . 1, 0

L (Σ0 )

Shock formation for quasilinear wave systems featuring multiple speeds 54

˘ [0,2] L X µ

L2 (Σ10 )

˘ [0,2] , X Lµ

L2 (Σ10 )

≤17;2 i

Z∗ L(Small) L∞ (Σ1 ) , Li(Small) 0

˘ [0,2]

LX µ

L∞ (Σ10 )

, kµ − 1kL2 (Σ1 ) , P∗[1,19] µ L2 (Σ1 ) . ˚ 0

0

˘ ˘

˘ [1,2] , XL Xµ , X µ . 1, 1 1 2 2 L (Σ0 )

2˚ δ−1 ∗ )

L∞ (P0

L∞ (Σ10 )

. 1,

[1,17]

P∗ µ ∞ 1 . ˚ kµ − 1kL∞ (Σ1 ) , kµ − 1k ∞ 2˚δ−1 , , L (Σ0 ) 0 L (P0 ∗ )

˘ [0,2]

˘ ˘

˘ [1,2] , X Lµ , XL Xµ , X µ . 1. 1 1 1 ∞ ∞ ∞ L (Σ0 )

L (Σ0 )

(3.4.2b)

L (Σ0 )

˘ [1,2] i

X L(Small)

.˚ ,

(3.4.2a)

(3.4.3) (3.4.4a) (3.4.4b)

L (Σ0 )

Proof. Readers may consult [28, Lemma 7.3] for a discussion of how to prove the estimates along Σ10 . The data in [28] were compactly supported in Σ10 , which is different than the present context, but that minor detail does not necessitate any substantial changes in the proof. ˚−1 To obtain the L∞ (P02δ∗ ) bounds for Li(Small) and µ − 1 stated in (3.4.3) and (3.4.4a), we first use the evolution equations (2.10.1)-(2.10.2), Lemma 2.11, and the fundamental theorem of calculus to deduce that for (t, ϑ) ∈ [0, 2˚ δ−1 ∗ ] × T, we have the following estimate, where f is smooth in its arguments: µ−1 (t, 0, ϑ) P2 (3.4.5) a a=1 |L(Small) | µ−1 (0, 0, ϑ) ≤ P2 a | |L a=1 (Small) Z t f(µ − 1, L1(Small) , L2(Small) , Ψ) Z ≤1 Ψ (s, 0, ϑ) ds. +C s=0

From the k · k -involving estimates stated in (3.4.3) and (3.4.4a), we see that the first ˚ term on RHS (3.4.5) is . n . Moreover, from (3.3.12), owe deduce that the time integral on Rt RHS (3.4.5) is ≤ C˚ s=0 f(µ − 1, L1(Small) , L2(Small) ) (s, 0, ϑ) ds. Hence, from Gronwall’s µ−1 (t, 0, ϑ) ≤ C˚ inequality, we conclude that P2 for t ∈ [0, 2˚ δ−1 a ∗ ], which in |L | a=1 (Small) particular implies the desired bound kµ − 1k ∞ 2˚δ−1 .˚ . ∗ L∞ (Σ10 )

L (P0

)

3.5. T(Boot) , the positivity of µ, and the diffeomorphism property of Υ. We now state some basic bootstrap assumptions. We start by fixing a real number T(Boot) with 0 < T(Boot) ≤ 2˚ δ−1 ∗ .

(3.5.1)

We assume that on the spacetime domain MT(Boot) ,U0 (see (1.7.3f)), we have µ > 0.

(BAµ > 0)

J. Speck

55

Inequality (BAµ > 0) implies that no shocks are present in MT(Boot) ,U0 . We also assume that The change of variables map Υ from Def. 2.18 is a C 1 diffeomorphism from

(3.5.2)

[0, T(Boot) ) × [0, U0 ] × T onto its image. 3.6. Fundamental L∞ bootstrap assumptions. Our fundamental bootstrap assump~ are that the following inequalities hold on MT tions for Ψ and W (see Subsect. 3.2 (Boot) ,U0 regarding the vectorfield operator notation):

≤11 ≤10 ~ ~)

P Ψ ∞ u , ≤ ε, (BAΨ − W

P W L (Σ ) u L∞ (Σt )

t

where ε is a small positive bootstrap parameter whose smallness we describe in Sect. 3.8. 3.7. Auxiliary L∞ bootstrap assumptions. In deriving pointwise estimates, we find it convenient to make the following auxiliary bootstrap assumptions. In Prop. 5.6, we will derive strict improvements of these assumptions. Auxiliary bootstrap assumptions for small quantities. We assume that the following inequalities hold on MT(Boot) ,U0 :

≤10;1

Z∗ Ψ L∞ (Σu ) ≤ ε1/2 , (AUX1Ψ) t

LP [1,9] µ ∞ u , P∗≤9 µ ∞ u ≤ ε1/2 , L (Σt ) L (Σt )

≤9;1 i

Z∗ L(Small) ≤ ε1/2 , L∞ (Σu t)

≤8;1 ≤ ε1/2 . /Z χ

L u ∞

(AUX1µ) (AUX1L(Small) ) (AUX1χ)

L (Σt )

Auxiliary bootstrap assumptions for quantities that are allowed to be large.

˘

˘ (AUX2Ψ) XΨ XΨ ≤

∞ u

∞ u + ε1/2 ,

L (Σ0 )

L (Σt )

1

˘

GLL XΨ

∞ u + ε1/2 , 2 L (Σ0 )

˘ ≤ 1 + 2˚ δ−1 GLL XΨ + ε1/2 ,

kLµkL∞ (Σut ) ≤

kµkL∞ (Σut ) ∗

˘ i

˘ i

XL(Small) ∞ u ≤ XL(Small) L (Σt )

L∞ (Σu 0)

L∞ (Σu 0)

+ ε1/2 .

(AUX2µ) (AUX3µ) (AUX2L(Small) )

3.8. Smallness assumptions. For the remainder of the article, when we say that “A is small relative to B,” we mean that there exists a continuous increasing function f : (0, ∞) → (0, ∞) such that A ≤ f (B). In principle, the functions f could always be chosen to be polynomials with positive coefficients or exponential functions. However, to avoid lengthening the paper, we typically do not specify the form of f . Throughout the rest of the paper, we make the following relative smallness assumptions. We continually adjust the required smallness in order to close our estimates.

Shock formation for quasilinear wave systems featuring multiple speeds 56 • The bootstrap parameter ε is small relative to ˚ δ−1 , where ˚ δ is the data-size parameter from Subsect. 3.3. • ε is small relative to51 the data-size parameter ˚ δ∗ from Def. 3.4. The first assumption will allow us to control error terms that, roughly speaking, are of size ε˚ δk for some integer k ≥ 0. The second assumption is relevant because the expected blowup-time is approximately ˚ δ−1 ∗ , and the assumption will allow us to show that various error products featuring a small factor ε remain small for t < 2˚ δ−1 ∗ , which is plenty of time for us to show that a shock forms. Finally, we assume that ˚ ≤ ε,

(3.8.1)

where ˚ is the data smallness parameter from from Subsect. 3.3 and Lemma 3.1. 4. Energies, Null Fluxes, and Energy-Flux Identities In this section, we define the energies and null fluxes that we use in our L2 analysis. We also provide the basic energy-null flux identities for solutions to the fast wave equation (1.6.2a) and to the slow wave equation, in the first-order form (1.6.11a)-(1.6.11d). 4.1. Definitions of the energies and null fluxes. 4.1.1. Energies and null fluxes for the fast wave. Definition 4.1 (Energy and null flux for the fast wave). In terms of the non-degenerate geometric forms of Def. 2.24, we define the fast wave energy functional E(F ast) [·] and fast wave null flux functional F(F ast) [·] as follows: Z 1 ˘ + 2(XΨ) ˘ 2 + 1 (1 + 2µ)µ|d/f |2 d$, E(F ast) [f ](t, u) = (1 + 2µ)µ(Lf )2 + 2µ(Lf )Xf 2 2 Σu t (4.1.1a) Z F(F ast) [f ](t, u) = (1 + µ)(Lf )2 + µ|d/f |2 d$. (4.1.1b) Put

4.1.2. Energies and null fluxes for the slow wave. Let V~ := (v, v0 , v1 , v2 )

(4.1.2)

be an array comprising four scalar functions. In later applications, the entries of V~ will be ~ defined in (1.6.1). To construct energies and null derivatives of the entries of the array W fluxes for the slow wave, we will rely on the compatible current vectorfield J = J[V~ ], which we define relative to the Cartesian coordinates as follows: J α [V~ ] := 2Qαβ [V~ ]δ 0 − v 2 (h−1 )αβ δ 0 . (4.1.3) β

β

In (4.1.3), 1 Qαβ [V~ ] := (h−1 )ακ (h−1 )βλ vκ vλ − (h−1 )αβ (h−1 )κλ vκ vλ 2 is the energy-momentum tensorfield of the slow wave equation (1.6.2b). 51Note

that ˚ δ and ˚ δ∗ are allowed to be small or large in an absolute sense.

(4.1.4)

J. Speck

57

Remark 4.1 (Suppression of some arguments of Q). Although Qαβ [V~ ] depends on ~ ) through the Cartesian component functions (h−1 )αβ = (h−1 )αβ (Ψ, W ~ ), we typically (Ψ, W suppress this dependence. Note that (1.6.13) and our assumption (g −1 )αβ (Ψ = 0) = (m−1 )αβ = diag(−1, 1, 1) to~ | is sufficiently small, the one-form with Cartesian compogether imply that when |γ| + |W 0 nents δα is past-directed (by (1.6.14)) and h-timelike. Thus, the first term on RHS (4.1.3) is the contraction of the energy-momentum tensorfield of the slow wave with a past-directed h-timelike one-form. We also note the following explicit formulas, the first of which relies on the second relation in (1.6.14), where (i = 1, 2): J 0 [V~ ] := 2(h−1 )α0 (h−1 )β0 vα vβ + (h−1 )αβ vα vβ + v 2 , J i [V~ ] := 2(h−1 )αi (h−1 )β0 vα vβ − (h−1 )i0 (h−1 )αβ vα vβ − (h−1 )i0 v 2 .

(4.1.5a) (4.1.5b)

Now, we again consider the past-directed one-form with Cartesian components δα0 , which ~ | is sufficiently small. Since this we have already shown to be h-timelike when |γ| + |W one-form is contracted on RHS (4.1.3), the well-known dominant energy condition for Q[V~ ] implies that if ω is any past-directed, h-timelike one-form and if J 6= 0, then J α ωα > 0. It follows that h(J, J) ≤ 0 and thus J is h-causal. Moreover, taking ωα := δα0 , we find that ~ | is sufficiently small: J 0 > 0. Thus, by (1.6.12), the following holds when |γ| + |W V~ 6= 0 =⇒ J is future-directed and g-timelike.

(4.1.6)

It is convenient to define the slow wave energy and null flux relative to the Cartesian volume forms. However, when describing their coerciveness properties and deriving energy estimates, we use the non-degenerate geometric forms; see Lemma 4.1. Definition 4.2 (Energy and null flux for the slow wave). Let J α [V~ ] be the compatible current vectorfield with Cartesian components given by (4.1.5a)-(4.1.5b). In terms of the Cartesian forms of Def. 2.26 and the one-form Hα defined in (2.17.1), we define the slow wave energy functional E(Slow) [·] and slow wave null flux functional F(Slow) [·] as follows: E(Slow) [V~ ](t, u) :=

Z Σu t

J [V~ ] dΣ, 0

F(Slow) [V~ ](t, u) :=

Z

J α [V~ ]Hα dP.

(4.1.7)

Put

4.2. Coerciveness of the energy and null flux for the slow wave. The coerciveness properties of the energy and null flux for the fast wave are fairly apparent from Def. 4.1. In contrast, it takes some effort to reveal the coerciveness of the energy and null flux for the slow wave. The next lemma yields the desired coerciveness. Lemma 4.1 (Coercivity of the energy and null flux for the slow wave). In terms of the non-degenerate geometric forms of Def. 2.24, the slow wave energy E(Slow) [·] and the slow wave null flux F(Slow) [·] from Def. 4.2 enjoy the following coerciveness properties, valid

Shock formation for quasilinear wave systems featuring multiple speeds 58 ~ | is sufficiently small: when |γ| + |W E(Slow) [V~ ](t, u) ≈ F(Slow) [V~ ](t, u) ≈

Z

( µ v2 +

2 X

Σu t

Z

) vα2

$,

(4.2.1a)

α=0

( v2 +

Put

2 X

) vα2

$.

(4.2.1b)

α=0

Remark 4.2 (On the necessity of the null fluxes and the necessity of the slow speed of the slow wave). It is critically important for our analysis that the null flux F(Slow) [V~ ] controls V~ without any degenerate factor of µ, as RHS (4.2.1b) shows. Similar remarks apply to the null flux F(F ast) [Ψ](t, u) defined in (4.1.1b). We also note that to derive ~ is slower than the wave Ψ, the estimate (4.2.1b), we need the assumption that the wave W and this is the main spot in the article where we use this assumption. Proof. Just above equation (4.1.6), we showed that J 0 = J 0 [V~ ] > 0 whenever V~ 6= 0 and ~ | is sufficiently small. Since J 0 [V~ ] is precisely quadratic in its arguments V~ , the |γ| + |W desired estimate (4.2.1a) follows from this fact, the second identity in (2.18.1), and the first definition in (4.1.7). To prove (4.2.1b), we first note that the one-form H defined in (2.17.1) is past-directed (in view of the last identity in (2.2.9a)) and g-null. Thus, by (4.1.6), J α Hα > 0 whenever ~ | is sufficiently small. Since J α Hα is precisely quadratic in V~ , the desired V~ 6= 0 and |γ| + |W estimate (4.2.1b) follows from the last identity in (2.18.1) and the second definition in (4.1.7). 4.3. Energy identities. In this subsection, we derive energy identities for solutions to the system (1.6.2a) + (1.6.11a)-(1.6.11d). 4.3.1. Energy identities for the fast wave. Proposition 4.2. [28, Proposition 3.5; Fundamental energy-flux identity for the fast wave] For solutions f to the inhomogeneous wave equation µg(Ψ) f = F, we have the following identity for t ≥ 0 and u ∈ [0, U0 ]: E(F ast) [f ](t, u) + F(F ast) [f ](t, u)

(4.3.1) Z

n o ˘ (1 + 2µ)(Lf ) + 2Xf F d$

= E(F ast) [f ](0, u) + F(F ast) [f ](t, 0) − Mt,u

−

1 2

Z Mt,u

[Lµ]− |d/f |2 d$ +

5 Z X i=1

Mt,u

(T )

P(i) [f ] d$,

J. Speck

59

where f+ := max{f, 0}, f− := max{−f, 0}, and 1 1 (T rans−Ψ) (T an−Ψ) (T ) 2 ˘ − µtrg/ χ − trg//k − µtrg//k , (4.3.2a) P(1) [f ] := (Lf ) − Lµ + Xµ 2 2 n o (T rans−Ψ) (T an−Ψ) (T ) ˘ P(2) [f ] := −(Lf )(Xf ) trg/ χ + 2trg//k + 2µtrg//k , (4.3.2b) ( ) ˘ 1 [Lµ] Xµ 1 + (T ) P(3) [f ] := µ|d/f |2 + + 2Lµ − trg/ χ − trg//k (T rans−Ψ) − µtrg//k (T an−Ψ) , 2 µ µ 2 (4.3.2c) (T ) (4.3.2d) P(4) [f ] := (Lf )(d/# f ) · (1 − 2µ)d/µ + 2ζ(T rans−Ψ) + 2µζ(T an−Ψ) , (T ) ˘ )(d/# f ) · d/µ + 2ζ(T rans−Ψ) + 2µζ(T an−Ψ) . (4.3.2e) P(5) [f ] := −2(Xf The tensorfields χ, ζ(T rans−Ψ) , /k (T rans−Ψ) , ζ(T an−Ψ) , and /k (T an−Ψ) from above are as in (2.4.4), (2.5.4a), (2.5.4b), (2.5.5a), and (2.5.5b), while the symbol “T ” in (4.3.2a)-(4.3.2e) merely signifies that the energies and error terms are tied to the multiplier vectorfield from (1.7.31), as is shown by the proof of [28, Proposition 3.5]. 4.3.2. Energy identities for the slow wave. We now derive the analog of Prop. 4.2 for the slow wave variable. Proposition 4.3 (Fundamental energy-flux identity for the slow wave). Solutions V~ to the inhomogeneous system µ∂t v0 = µ(h−1 )ab ∂a vb + 2µ(h−1 )0a ∂a v0 + F0 ,

(4.3.3a)

µ∂t vi = µ∂i v0 + Fi ,

(4.3.3b)

µ∂t v = µv0 + F,

(4.3.3c)

µ∂i vj = µ∂j vi + Fij

(4.3.3d)

verify the following energy identity for t ≥ 0 and u ∈ [0, U0 ]: E(Slow) [V~ ](t, u) + F(Slow) [V~ ](t, u) (4.3.4) = E(Slow) [V~ ](0, u) + F(Slow) [V~ ](t, 0) Z + {1 + γf(γ)} W[V~ ] d$ Mt,u Z + {1 + γf(γ)} 4(h−1 )α0 (h−1 )β0 vα Fβ + 2(h−1 )αβ vα Fβ + 2(h−1 )αa (h−1 )b0 vα Fab + 2vF d$, Mt,u

where the schematically denoted function γf(γ) is smooth and vanishes when γ = 0, and W[V~ ] := 4µ(∂t (h−1 )α0 )(h−1 )β0 vα vβ + µ(∂t (h−1 )αβ )vα vβ

(4.3.5)

+ 2µ(∂a (h−1 )αa )(h−1 )β0 vα vβ + 2µ(h−1 )αa (∂a (h−1 )β0 )vα vβ − µ(∂a (h−1 )a0 )(h−1 )αβ vα vβ − µ(h−1 )a0 (∂a (h−1 )αβ )vα vβ − µ(∂a (h−1 )a0 )v 2 − 2µ(h−1 )α0 vvα .

Shock formation for quasilinear wave systems featuring multiple speeds 60 Proof. We consider the vectorfield J = J[V~ ] defined relative to the Cartesian coordinates by (4.1.3). Using the second relation in (1.6.14), the identities (4.1.5a)-(4.1.5b), and equations (4.3.3a)-(4.3.3d), we compute that ∂α J α = 4(∂t (h−1 )α0 )(h−1 )β0 vα vβ + (∂t (h−1 )αβ )vα vβ −1 αa

−1 β0

−1 αa

(4.3.6)

−1 β0

+ 2(∂a (h ) )(h ) vα vβ + 2(h ) (∂a (h ) )vα vβ − (∂a (h−1 )a0 )(h−1 )αβ vα vβ − (h−1 )a0 (∂a (h−1 )αβ )vα vβ − (∂a (h−1 )a0 )v 2 − 2(h−1 )α0 vvα 4 2 2 2 + (h−1 )α0 (h−1 )β0 Fα vβ + (h−1 )αβ Fα vβ + (h−1 )αa (h−1 )b0 vα Fab + vF. µ µ µ µ We now apply the divergence theorem to the vectorfield J on the region Mt,u , where we use the Cartesian coordinates, the Euclidean metric δ αβ := diag(1, 1, 1) on R × Σ, and the Euclidean forms of Def. 2.26 in all computations. As the final step, we use the identities (2.18.1) to express all integrals relative to the geometric forms of Def. 2.24. Also taking into account Def. 4.2, we arrive at the desired identity (4.3.4). Note that the one-form Hα on RHS (4.1.7) is the Euclidean unit-length co-normal to the hypersurfaces Pu , which is the reason that J α [V~ ]Hα arises when we apply the standard Euclidean divergence theorem on Mt,u . 5. Preliminary pointwise estimates In this section, we use the data-size and bootstrap assumptions of Sect. 3 to derive preliminary L∞ and pointwise estimates for the solution. These estimates serve as the starting point for related estimates that we derive in Sects. 6-8 Remark 5.1 (Many estimates were proved in [28]). Many of the estimates involving the fast wave variable Ψ and the eikonal function are independent of the slow wave variable ~ and were proved in [28]; thus, we cite [28] whenever possible. For a few of the estimates W involving the non-zero data along P0 , we instead cite [16] since, unlike in [28], the data were allowed to be non-zero along P0 in [16], as in the present article. 5.1. Notation for repeated differentiation. Definition 5.1 (Notation for repeated differentiation). Recall that the commutation vectorfield sets Z and P are defined in Def. 2.20. We label the three vectorfields in Z as ˘ Note that P = {Z(1) , Z(2) }. We define the following follows: Z(1) = L, Z(2) = Y, Z(3) = X. vectorfield operators: ~ := N with ι1 , ι2 , · · · , ιN ∈ {1, 2, 3}, • If I~ = (ι1 , ι2 , · · · , ιN ) is a multi-index of order |I| ~

then Z I := Z(ι1 ) Z(ι2 ) · · · Z(ιN ) denotes the corresponding N th order differential oper~ ator. We write Z N rather than Z I when we are not concerned with the structure ~ of I. ~ • Similarly, L /IZ := L /Z(ι ) L /Z(ι ) · · · L /Z(ι ) denotes an N th order `t,u -projected Lie deriva1

2

N

tive operator (see Def. 2.10), and we write L /N Z when we are not concerned with the ~ structure of I.

J. Speck

61

• If I~ = (ι1 , ι2 , · · · , ιN ), then I~1 + I~2 = I~ means that I~1 = (ιk1 , ιk2 , · · · , ιkm ) and I~2 = (ιkm+1 , ιkm+2 , · · · , ιkN ), where 1 ≤ m ≤ N and k1 , k2 , · · · , kN is a permutation of 1, 2, · · · , N . • Sums such as I~1 + I~2 + · · · + I~M = I~ have an analogous meaning. ~ • Pu -tangent operators such as P I are defined analogously, except in this case we clearly have ι1 , ι2 , · · · , ιN ∈ {1, 2}. 5.2. Basic assumptions, facts, and estimates that we use silently. For the reader’s convenience, we present here some basic assumptions, facts, and estimates (similar to those from [28, Section 8.2]) that we silently use throughout the rest of the paper when deriving estimates. (1) All of the estimates that we derive hold on the bootstrap region MT(Boot) ,U0 . Moreover, in deriving estimates, we rely on the data-size and bootstrap assumptions of Subsects. 3.5-3.7, the smallness assumptions of Subsect. 3.8, and the estimates for the data of the eikonal function quantities provided by Lemma 3.1. (2) All quantities that we estimate can be controlled in terms of the quantities γ = ~ , and their derivatives (where the X ˘ derivatives do {Ψ, µ − 1, L1(Small) , L2(Small) }, W not have to be small, nor do µ − 1 or Lµ). (3) We typically use the Leibniz rule for the operators L /Z and ∇ / when deriving pointwise estimates for the L /Z and ∇ / derivatives of tensor products of the schematic form Qm v , where the v are scalar functions or `t,u -tangent tensors. Our derivative i i=1 i counts are such that all vi except at most one are uniformly bounded in L∞ on MT(Boot) ,U0 . Thus, our pointwise estimates often explicitly feature (on the right-hand sides) only the factor with the most derivatives on it, multiplied by a constant (often implicit) that uniformly bounds the other factors. In some estimates, the right-hand sides also gain a smallness factor, such as ε1/2 , generated by the remaining vi0 s. /, as shown by Lemma 2.6. (4) The operators L /N Z commute through d (5) As differential operators acting on scalar functions, we have Y = (1 + O(γ)) d / = 1/2 (1 + O(ε ))d/, a fact which follows from the proof of [28, Lemma 8.2] and the bootstrap assumptions. Hence, for scalar functions f , we sometimes schematically depict d/f as (1 + O(γ)) P f or P f when the factor 1+O(γ) is not important. Similarly, from Lemma 2.11 and the proofs of [28, Lemma 8.2] and [28, Lemma 8.3], we find that we [1,2] [1,2] ≤1 can depict ∆ / f by52 f(P γ)P∗ f (or P∗ f when the factor f(P ≤1 γ) is not impor tant) and, for type n0 `t,u -tangent tensorfields ξ, ∇ / ξ by f(P ≤1 γ, g/−1 , d/x1 , d/x2 )L /≤1 P ξ ≤1 53 ≤1 −1 1 2 (or L /P ξ when the factor f(P γ, g/ , d/x , d/x ) is not important ). (6) The constants in all of our estimates are allowed to depend on the data-size parameters ˚ δ and ˚ δ−1 ∗ . 5.3. Omission of the independent variables in some expressions. We use the following notational conventions in the rest of the article. 52In [28],

[1,2]

we schematically denoted ∆ / f by f(P ≤1 γ, g/−1 )P∗ f . Here we note that in fact, the dependence on f on g/ is not needed. 53In the analogous discussion in [28], the dependence of f on d /x1 , d/x2 was mistakenly omitted. −1

Shock formation for quasilinear wave systems featuring multiple speeds 62 • Many of our pointwise estimates are stated in the form |f1 | . F (t, u)|f2 |

• • • •

for some function F . Unless we otherwise indicate, it is understood that both f1 and f2 are evaluated at the point with geometric R coordinates (t, u, ϑ). Unless we otherwise indicate, in integrals `t,u f dλg/ , the integrand f and the length form dλg/ are viewed as functions of (t, u,Rϑ) and ϑ is the integration variable. Unless we otherwise indicate, in integrals Σu f d$, the integrand f and the area form t 0 d$ are viewed as functions of (t, u0 , ϑ) and R (u , ϑ) are the integration variables. Unless we otherwise indicate, in integrals P t f d$, the integrand f and the area form u d$ are viewed as functions of (t0 , u, ϑ) and R(t0 , ϑ) are the integration variables. Unless we otherwise indicate, in integrals Mt,u f d$, the integrand f and the volume form d$ are viewed as functions of (t0 , u0 , ϑ) and (t0 , u0 , ϑ) are the integration variables.

5.4. Differential operator comparison estimates. Lemma 5.1. [28, Lemma 8.2; Controlling ∇ / derivatives in terms of Y derivatives] Let f be a scalar function on `t,u . Then the following estimates hold:54 |d/f | ≤ (1 + Cε1/2 ) |Y f | ,

|∇ / 2 f | ≤ (1 + Cε1/2 ) |d/(Y f )| + Cε1/2 |d/f |.

(5.4.1)

5.5. Pointwise estimates for Cartesian components and for the Lie derivatives of the metric on `t,u . Lemma 5.2. [28, Lemma 8.4; Pointwise estimates for xi ] Assume that N ≤ 18. Let xi = xi (t, u, ϑ) denote the Cartesian coordinate function and let ˚ xi = ˚ xi (u, ϑ) := xi (0, u, ϑ). Then the following estimates hold for i = 1, 2 (see Subsect. 3.2 regarding the vectorfield operator notation):55 i x − ˚ (5.5.1a) xi . 1, i d/x . 1 + |γ| , (5.5.1b) d/P [1,N ] xi . P ≤N γ , (5.5.1c) d/Z [1,N ];1 xi . Z∗≤N ;1 γ + P∗[1,N ] γ . (5.5.1d) In the case i = 2 at fixed u, ϑ, LHS (5.5.1a) is to be interpreted as the Euclidean distance traveled by the point x2 in the flat universal covering space R of T along the corresponding integral curve of L over the time interval [0, t]. Lemma 5.3. [28, Lemma 8.4; Crude pointwise estimates for the Lie derivatives of g/ and g/−1 ] Assume that N ≤ 18. Then the following estimates hold (see Subsect. 3.2 regarding 54We

have corrected a minor typo from [28]: the second factor of “ε1/2 ” on RHS (5.4.1) was incorrectly listed as “ε” in [28, Equation (8.2)]. 55The estimate (5.5.1d) is correct as stated, which fixes a minor typo from [28, Equation (8.11e)].

J. Speck the vectorfield operator notation): N +1 N +1 −1 N N /P g/ , L /P g/ L /P χ , P trg/ χ . P ≤N +1 γ , L N +1;1 N +1;1 −1 N ;1 N ;1 L / g / , L / g / , L / χ Z∗ Z∗ Z , Z trg/ χ . Z∗≤N +1;1 γ + P∗[1,N +1] γ , N +1;1 N +1;1 −1 ≤N +1;1 [1,N +1] /Z g/ , L /Z g/ . Z γ + P∗ γ . L

63

(5.5.2a) (5.5.2b) (5.5.2c)

5.6. Commutator estimates. In this subsection, we establish some commutator estimates. Lemma 5.4. [28, Lemma 8.7; Pure Pu -tangent commutator estimates] Assume that ~ = N + 1 multi-index for the set P of Pu -tangent 1 ≤ N ≤ 18. Let I~ be an order |I| ~ Let f be a scalar commutation vectorfields (see Def. 5.1), and let I~0 be any permutation of I. 0 function, and let ξ be an `t,u -tangent one-form or a type 2 `t,u -tangent tensorfield. Then the following commutator estimates hold: ~0 I~ (5.6.1a) P f − P I f . ε1/2 P∗[1,N ] f + P∗[1,bN/2c] f P ≤N γ . Moreover, if 1 ≤ N ≤ 17 and I~ is as above, then the following commutator estimates hold: 2 [∇ / , P N ]f . ε1/2 P∗[1,N ] f + P∗[1,dN/2e] f P ≤N +1 γ , [∆ / , P N ]f . ε1/2 P∗[1,N +1] f + P∗[1,dN/2e] f P ≤N +1 γ , ~ ≤bN/2c ≤N +1 I I~0 1/2 [1,N ] ξ γ , ξ + L / ξ . ε L / ξ − L / L / P P P P P ≤bN/2c ≤N +1 1/2 ≤[1,N −1] ξ . ε L / /,L /N ]ξ ξ P γ , + /P [∇ L P P | {z } Absent if N = 1 ≤bN/2c ≤N +1 1/2 [1,N ] ξ γ . ξ + L / /,L /N ]ξ . ε L / P [div P P P

(5.6.2a) (5.6.2b)

(5.6.3a) (5.6.3b)

(5.6.3c)

Finally, if 1 ≤ N ≤ 17, then we have the following alternate version of (5.6.2a): 2 [∇ (5.6.4) / , P N ]f . P ≤dN/2e+1 γ P∗[1,N ] f + P∗[1,dN/2e] f P ≤N +1 γ . Lemma 5.5. [28, Lemma 8.8; Mixed Pu -transversal-tangent commutator estimates] ~ Assume that 1 ≤ N ≤ 18. Let Z I be a Z -multi-indexed operator containing exactly ˘ factor, and assume that |I| ~ = N + 1. Let I~0 be any permutation of I. ~ Let f be a one X scalar function. Then the following commutator estimates hold (see Subsect. 3.2 regarding the vectorfield operator notation): I~ I~0 (5.6.5) Z f − Z f . P∗[1,N ] f + ε1/2 Y Z ≤N −1;1 f | {z } Absent if N = 1 [1,bN/2c] P∗[1,N ] γ + Y Z ≤bN/2c−1;1 f P ≤N γ . + P∗ f Z∗≤N ;1 γ | {z } Absent if N ≤ 3

Shock formation for quasilinear wave systems featuring multiple speeds 64 Moreover, if 1 ≤ N ≤ 17, then the following estimates hold: 2 [∇ / , Z N ;1 ]f . Z∗≤N ;1 f (5.6.6a) P∗[1,N +1] γ ≤dN/2e ≤N +1 + Z∗ f P γ , + P ≤dN/2e f Z∗≤N +1;1 γ [∆ (5.6.6b) / , Z N ;1 ]f . Z∗≤N +1;1 f [1,N +1] P∗ γ ≤dN/2e ≤N +1 + P ≤dN/2e f + Z∗ f P γ . ≤N +1;1 Z γ ∗

5.7. Transport inequalities and improvements of the auxiliary bootstrap assumptions. In this subsection, we use the previous estimates to derive transport inequalities for the eikonal function quantities and improvements of the auxiliary bootstrap assumptions. The transport inequalities form the starting point for our derivation of L2 estimates for the below-top-order derivatives of the eikonal function quantities as well as their top-order derivatives involving at least one L differentiation (see Lemma 9.12). The main challenge in proving the proposition is to propagate the smallness of the ˚ −sized quantities even ˚ though some terms in the evolution equations involve δ−sized quantities, which are allowed to be large. To this end, we must find and exploit effective partial decoupling between various quantities, which is present because of the special structure of the evolution equations relative to the geometric coordinates, because of our assumptions on the structure of the inhomogeneous terms in the wave equations (including (1.6.5)), and because of the good properties of the commutation vectorfield sets Z and P. Proposition 5.6 (Transport inequalities and improvements of the auxiliary bootstrap assumptions). The following estimates hold (see Subsect. 3.2 regarding the vectorfield operator notation): Transport inequalities for the eikonal function quantities. •Transport inequalities for µ. The following pointwise estimate holds: |Lµ| . Z ≤1 Ψ . Moreover, for 1 ≤ N ≤ 18, the following estimates hold: LP N µ , P N Lµ . Z∗≤N +1;1 Ψ + P ≤N γ + ε P∗[1,N ] γ .

(5.7.1a)

(5.7.1b)

•Transport inequalities for Li(Small) and trg/ χ. For N ≤ 18, the following estimates hold: LP N Li(Small) P N LLi(Small) ≤N +1 , . P + ε P ≤N γ , Ψ (5.7.2a) N −1 N −1 P LP trg/ χ Ltrg/ χ LZ N ;1 Li(Small) Z N ;1 LLi(Small) ≤N +1;1 εP∗[1,N ] γ , . Z∗ . (5.7.2b) Ψ + LZ N −1;1 trg/ χ Z N −1;1 Ltrg/ χ Z∗≤N ;1 γ ~ , and the eikonal function quantities. L∞ estimates for Ψ, W

J. Speck

65

•L∞ estimates involving at most one transversal derivative of Ψ. The following estimates hold:

˘

˘ (5.7.3a)

XΨ ∞ u ≤ XΨ ∞ u + Cε, L (Σt ) L (Σ0 )

≤10;1

Z∗ Ψ L∞ (Σu ) ≤ Cε. (5.7.3b) t

~ . The following •L estimates involving at most one transversal derivative of W estimates hold:

≤10;1 ~ W ≤ Cε. (5.7.4)

Z u ∞

L∞ (Σt )

∞

•L

estimates for µ. The following estimates hold:

1

˘ + O(ε), kLµkL∞ (Σut ) = GLL XΨ 2 L∞ (Σu 0)

LP [1,9] µ ∞ u , P∗[1,9] µ ∞ u ≤ Cε,

(5.7.5b)

˘ δ−1 G XΨ kµ − 1kL∞ (Σut ) ≤ 2˚

∗ LL

(5.7.6a)

L (Σt )

L (Σt )

L∞ (Σu 0)

+ Cε.

•L∞ estimates for Li(Small) and χ. The following estimates hold:

≤10 i

LP ≤10 Li(Small)

, P L (Small) L∞ (Σu ) ≤ Cε, L∞ (Σu t) t

≤9;1 i

LZ ≤9;1 Li(Small) , Z∗ L(Small) L∞ (Σu ) ≤ Cε, L∞ (Σu t) t

˘ i

˘ i

≤ XL

XL(Small)

(Small) u

(5.7.7a) (5.7.7b)

L∞ (Σu 0)

L∞ (Σt )

+ Cε,

≤9

≤9 #

≤9

P trg/ χ ∞ u ≤ Cε, χ , L / , /P χ

L P L (Σt ) L∞ (Σu L∞ (Σu t) t)

≤8;1

≤8;1 #

≤8;1

Z

∞ u ≤ Cε. χ , tr χ L / , /Z χ

L g / Z L (Σ ) u u ∞ ∞ L (Σt )

L (Σt )

(5.7.5a)

(5.7.7c) (5.7.8a) (5.7.8b)

t

Proof sketch. To derive (5.7.4) in the case Z ≤10;1 = P 10 , we simply note that the desired ~ ). To prove (5.7.4) in the bound is one of the bootstrap assumptions from (BAΨ − W ≤10;1 ˘ we first apply P ≤9 to the identity remaining case in which Z contains a factor of X,

≤9 ˘ ~ (2.15.1) and use the bootstrap assumptions to deduce that P X W . ε. We then u L∞ (Σt )

~ , the estimate just proved for P ≤9 X ˘W ~, use the commutator estimate (5.6.5) with f = W and the bootstrap assumptions, which allow us to arbitrarily commute the vectorfields in the ˘W ~ up to errors bounded in k · kL∞ (Σu ) by . ε. In total, we have derived expression P ≤9 X t

~ the desired bound Z ≤10;1 W . ε.

L∞ (Σu t)

The remaining estimates in Prop. 5.6 can be established using arguments nearly identical to the ones used in proving [28, Proposition 8.10], as we now outline. Specifically, one uses the transport equations of Lemma 2.8, the estimates of Lemma 3.1 and Lemmas 5.2-5.3, and the commutator estimates of Subsect. 5.6 to derive the desired bounds for µ and Li(Small) ;

Shock formation for quasilinear wave systems featuring multiple speeds 66 ~ and hence the proofs from [28, Proposition 8.10] these bounds are not explicitly tied to W go through verbatim. The estimates for χ then follow from these estimates with the help of Lemmas 2.10 and 2.11. To derive the desired estimate (5.7.3a) for Ψ and the estimate ˘ (the desired bounds in the case (5.7.3b) for Ψ when Z∗≤10;1 contains exactly one factor of X ≤10;1 ≤10 ~ )), case Z∗ =P are restatements of one of the bootstrap assumptions (BAΨ − W one can use use equation (2.13.1a), equation (2.10.1), Lemma 2.11, Lemma 2.13, and the assumptions (1.6.5) on the semilinear terms to rewrite the wave equation (1.6.2a) for Ψ in the following schematic “transport equation” form: ˘ = f(γ)∆ ˘ ~ g/−1 , d/x1 , d/x2 , P Ψ, XΨ)P ˘ LXΨ / Ψ + f(γ, g/−1 , d/x1 , d/x2 , P Ψ, XΨ)P P Ψ + f(γ, W, γ (5.7.9) ~ XΨ, ˘ P Ψ)Ψ + f(γ, W, ~ P Ψ, XΨ) ˘ W. ~ + f(γ, W, Then by applying P ≤9 to (5.7.9) and using the the bootstrap assumptions, one can show that

commutator estimates of Subsect. 5.6 and ˘ . ε, from which the bound LP ≤9 XΨ ˘ L∞ (Σu ) . ε easily follows by integrating in time (recall that L = ∂ ) and uskP ≤9 XΨk t ∂t ing the smallness assumptions on the data along Σ10 . Then by further applications of the commutator estimates of Subsect. 5.6, we obtain kZ∗≤10;1 ΨkL∞ (Σut ) . ε. More precisely, all terms that arise from differentiating RHS (5.7.9) with P ≤9 were handled in the proof of ~ . Note in particular that the commu[28, Proposition 8.10] except for the ones involving W ≤9 tator estimates needed to commute P through the operator L on LHS (5.7.9) and through ~ in any way. That is, the only influence the operator ∆ / on RHS (5.7.9) do not involve W ~ on the estimates under consideration is through the terms P ≤9 W ~ that arise from of W ~ kL∞ (Σu ) ≤ ε stated in (BAΨ − W ~ ), RHS (5.7.9). Due to the bootstrap assumption kP ≤10 W t ~ make only a negligible O(ε) contribution to the the products containing a factor of P ≤9 W estimates. For this reason, the same bounds stated in [28, Proposition 8.10] also hold in the present context. The following corollary is an immediate consequence of the fact that we have improved the auxiliary bootstrap assumptions of Subsect. 3.7 by showing that they hold with ε1/2 replaced by Cε. Corollary 5.7 (ε1/2 can be replaced by Cε). All prior inequalities whose right-hand sides feature an explicit factor of ε1/2 remain true with ε1/2 replaced by Cε. In particular, the auxiliary bootstrap assumptions of Subsect. 3.7 hold with ε1/2 replaced by Cε. 6. L∞ Estimates Involving Higher Transversal Derivatives

[Xµ]

C

˘ + Our energy estimates rely on the delicate estimate ≤ p (see

µ ∞ u T(Boot) − t L (Σt )

˘ ˘ (7.1.9)), whose proof relies on the bound X Xµ . 1. In this section, we derive this u L∞ (Σt )

bound and related ones that are needed to prove it.

J. Speck

67

6.1. Commutator estimates involving two transversal derivatives. We start with ˘ some basic commutator estimates involving two factors of the Pu -transversal vectorfield X. Lemma 6.1. [28, Lemma 9.1; Mixed Pu -transversal-tangent commutator estimates ˘ derivatives] Let Z I~ be a Z -multi-indexed operator containing exactly involving two X ˘ factors and let I~0 be any permutation of I. ~ If |I| ~ = 4, then we have the following two X commutator estimates: ( ) 2 X 0 ~ I~ ≤2 ˘ ≤2;1 ˘ XΨ| ˘ + ˘ XL ˘ a |P ≤1 X | + |P Xµ| + 1 Y Z f Z f − Z I f . |P ≤1 X (Small) a=1

(6.1.1a) + ε Y Z ≤2;2 f . ~ = 3, then Moreover, if |I| n o I~ I~0 ≤1 ˘ Y Z ≤1 f . f − Z f . |P Xµ| + 1 Z Finally, we have ( 2 X ≤1 ˘ ˘ ˘ XL ˘ a ˘ ˘ |P ≤1 X . |P X XΨ| + [∆ / , X X]f

(Small) |

(6.1.1b)

) ˘ + 1 Y Z ≤2;1 f . + |P ≤2 Xµ|

a=1

(6.1.2) Proof sketch. The estimates (6.1.1a)-(6.1.1b) and (6.1.2) were essentially proved as [28, Lemma 9.1]. Let us simply make a few clarifying remarks about the estimate (6.1.1a); similar remarks apply for the estimates (6.1.1b) and (6.1.2). Specifically, if one examines the proof of [28, Lemma 9.1], one finds that the estimate (6.1.1a) holds with the term in braces on LHS (6.1.1a) replaced by Z ≤3;2 γ + Z ≤3;1 γ . Using the L∞ estimates of Prop. 5.6, P we can bound these terms by . |Z ≤3;2 Ψ| + 2a=1 |Z ≤3;2 La(Small) | + |Z ≤3;1 µ| + 1, where the ˘ and the operators Z ≤3;1 contain exactly operators Z ≤3;2 contain exactly two factors of X ˘ Then by further commutations of the type used in proving [28, Lemma 9.1], one factor of X. one can ensure that up to error terms accounted for in the term in braces on LHS (6.1.1a), ˘ act first, which yields (6.1.1a). the factors of X We close this proof sketch by noting that in [28, Lemma 9.1], the authors were able to bound the terms in braces on RHSs (6.1.1a)-(6.1.1b) and (6.1.2) by . 1 by relying on additional “auxiliary” bootstrap assumptions; we do not make such auxiliary bootstrap assumptions in the present paper because our wave equation (1.6.2a) features new kinds of inhomogeneous terms that obstruct using a bootstrap approach of the type used in [28] in proving the analogous estimates. 6.2. The main estimates involving higher-order transversal derivatives. We now prove the main result of Sect. 6. Proposition 6.2 (L∞ estimates involving higher-order transversal derivatives). We have the following estimates.

Shock formation for quasilinear wave systems featuring multiple speeds 68 L∞ estimates involving two or three transversal derivatives of Ψ.

LZ ≤4;2 Ψ ∞ u . 1, L (Σt )

≤4;2

Z

Ψ L∞ (Σu ) . 1, t

˘ ˘ ˘

LX X XΨ ∞ u . 1, L (Σt )

˘ ˘ ˘ . 1.

X X XΨ u

(6.2.1b)

L∞ estimates involving one or two transversal derivatives of µ.

LZ ≤3;1 µ ∞ u . 1, L (Σt )

≤3;1

Z

µ L∞ (Σu ) . 1.

(6.2.2a)

L∞ (Σt )

(6.2.1a)

(6.2.1c) (6.2.1d)

(6.2.2b)

t

˘ ˘

LX Xµ ∞ u . 1, L (Σt )

˘ ˘

X Xµ ∞ u . 1.

(6.2.2c) (6.2.2d)

L (Σt )

L∞ estimates involving two transversal derivatives of Li(Small) .

≤3;2 i

Z L(Small) L∞ (Σu ) . 1.

(6.2.3)

t

~. L∞ estimates involving two transversal derivatives of W

≤3;2 ~ Z W

∞ u . 1.

(6.2.4)

L (Σt )

Sharp pointwise estimates involving the critical factor GLL . If 0 ≤ s ≤ t < T(Boot) , then we have the following estimates: |GLL (t, u, ϑ) − GLL (s, u, ϑ)| ≤ Cε(t − s), ˘ ˘ u, ϑ) − [GLL XΨ](s, u, ϑ) ≤ Cε(t − s). [GLL XΨ](t,

(6.2.5) (6.2.6)

Furthermore, with L(F lat) = ∂t + ∂1 , we have 1 ˘ u, ϑ) + O(ε), Lµ(t, u, ϑ) = GL(F lat) L(F lat) (Ψ = 0)XΨ(t, 2

(6.2.7)

where GL(F lat) L(F lat) (Ψ = 0) is a non-zero constant (see (1.6.9)). Proof. See Subsect. 5.2 for some comments on the analysis. Throughout this proof, we refer to the bounds of Subsect. 3.3 and Lemma 3.1 as the “conditions on the data.” Proof of (6.2.1a)-(6.2.1b), (6.2.2a)-(6.2.2b), and (6.2.3):

J. Speck

69

The main step is to obtain the following system of inequalities: 2 X ≤3;2 a Z L(Small) . Z ≤3;2 Ψ + Z ≤2;1 µ + 1,

(6.2.8)

a=1

˘ XΨ ˘ . Z ≤4;2 Ψ + Z ≤3;1 γ + 1 LP ≤2 X ≤2 ˘ ˘ ≤3;2 a . P X XΨ + Z L(Small) + Z ≤3;1 µ + 1, 2 2 X X ≤3;2 a ˘ XΨ ˘ + LZ ≤3;1 µ . P ≤2 X Z L(Small) + Z ≤3;1 µ + 1. a=1

(6.2.9)

(6.2.10)

a=1

Once we have obtained (6.2.8)-(6.2.10), we substitute RHS (6.2.8) for the term Z ≤3;2 La(Small) on RHSs (6.2.9), and (6.2.10). Then from the conditions on the data along Σ10 and Gronwall’s ∂ inequality (recall that L = ), we deduce that ∂t 2 ≤2 ˘ ˘ X ≤3;2 a Z L(Small) + Z ≤3;1 µ . 1. (6.2.11) P X XΨ + a=1

Moreover, using the commutator estimate (6.1.1a) with f = Ψ, the L∞ estimates of Prop. 5.6, ˘ XΨ ˘ and the estimate (6.2.11), we can arbitrarily permute the vectorfield factors in P ≤2 X up to errors bounded in k · kL∞ (Σut ) by . 1, which implies that ≤4;2 Z Ψ . 1. (6.2.12) All of the bounds under consideration now follow from (6.2.8)-(6.2.12). It remains for us to prove (6.2.8)-(6.2.10). To prove (6.2.8), we first note that by the L∞ estimates of Prop. 5.6, it suffices to consider the case in which the operator Z ≤3;2 contains ˘ Thus, to proceed, we write (2.10.3) in the following schematic form exactly two factors of X. with the help of Lemma 2.11: ˘ i ˘ P µ). XL /−1 , d/x1 , d/x2 , P Ψ, XΨ, (Small) = f(γ, g

(6.2.13)

˘ to (6.2.13) and using Lemmas 5.2 and 5.3 and the L∞ estimates of Prop. 5.6, Applying P ≤1 X ˘ XL ˘ i we deduce that P ≤1 X (Small) . RHS (6.2.8). Then using the commutator estimate (6.1.1b) with f = Li(Small) and the L∞ estimates of Prop. 5.6, we can arbitrarily permute ˘ XL ˘ i the vectorfield factors in the expression P ≤1 X (Small) up to errors that are bounded by ≤ RHS (6.2.8), which yields the desired bound (6.2.8). ˘ and The bound (6.2.9) follows from commuting the wave equation (5.7.9) with P ≤2 X using Lemmas 5.2 and 5.3, the commutator estimate (5.6.6b) with f = Ψ (to commute ˘ through ∆ ˘ (to commute P ≤2 X ˘ P ≤2 X / ), the commutator estimate (5.6.5) with f = XΨ through the operator L on LHS (5.7.9)), the commutator estimate (6.1.1a) with f = Ψ (to pass to the second line of RHS (6.2.9)), and the L∞ estimates of Prop. 5.6; the key point is that in Prop. 5.6, we already bounded all terms in the norm k · kL∞ (Σut ) by . 1, except for the terms featured on RHS (6.2.9), which appear only linearly.

Shock formation for quasilinear wave systems featuring multiple speeds 70 To derive (6.2.10), we first note that by the L∞ estimates of Prop. 5.6, it suffices to ˘ To proceed, consider the case in which the operator Z ≤3;1 contains exactly one factor of X. we write (2.10.1) in the following schematic form with the help of Lemma 2.11: ˘ + f(γ)P Ψ. Lµ = f(γ)XΨ

(6.2.14)

We then commute equation (6.2.14) with Z ≤3;1 and argue as in the proof of (6.2.9), where we handle the commutator term [L, Z ≤3;1 ]µ via the estimate (5.6.5) with f = µ; these steps yield (6.2.10).

~ Proof of (6.2.4): We first note that by (5.7.4), it suffices to show that Z ≤3;2 W

∞ u .1 L (Σt )

˘ To proceed, we apply P ≤1 X ˘ to the whenever Z contains precisely two factors of X. ∞ identity (2.15.1). Using the L estimates of Prop.

5.6 and the already proven estimates

≤1 ˘ ˘ ~ (6.2.1b) and (6.2.2b), we deduce that P X X W . 1. Then using the commutator u ≤3;2

L∞ (Σt )

~ , the L∞ estimates of Prop. 5.6, and the already proven estimate estimate (6.1.1b) with f = W ˘X ˘W ~ up (6.2.2b), we can arbitrarily permute the vectorfield factors in the expression P ≤1 X to errors bounded in k · kL∞ (Σut ) by . 1, thereby obtaining the desired bound. Proof of (6.2.1c)-(6.2.1d) and (6.2.2c)-(6.2.2d): The main step is to obtain the following system of inequalities: ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ L X X XΨ . X X XΨ + X Xµ (6.2.15) + 1, ˘ ˘ ˘ ˘ ˘ ˘ ˘ (6.2.16) LX Xµ . X X XΨ + X Xµ + 1. Then from (6.2.15)-(6.2.16), the conditions on the data along Σ10 , and Gronwall’s inequality, we deduce that ˘ ˘ ˘ ˘ ˘ (6.2.17) X X XΨ + X Xµ . 1. All four of the desired bounds now follow from (6.2.15)-(6.2.17). It remains for us to prove (6.2.15)-(6.2.16). To prove (6.2.15), we commute the wave equa˘ X. ˘ Our argument relies on the estimates |L tion (5.7.9) with X /X˘ X˘ g/−1 | . 1 and |L /X˘ X˘ d/xi | . 1, which we need to bound the L /X˘ X˘ derivatives of the factors g/−1 and d/xi on RHS (5.7.9). We will derive these estimates at the end of the proof using an argument that is independent of the estimates (6.2.1c)-(6.2.1d) and (6.2.2c)-(6.2.2d) under consideration. Combining the estimates |L /X˘ X˘ g/−1 | . 1 and |L /X˘ X˘ d/xi | . 1 with Lemmas 5.2 and 5.3, the commutator esti˘ (to commute X ˘X ˘ through the vectorfield L on LHS (5.7.9)), mate (6.1.1b) with f = XΨ ˘X ˘ through the operator ∆ the commutator estimate (6.1.2) with f = Ψ (to commute X / ∞ on RHS (5.7.9)), the L estimates of Prop. 5.6, and the already proven estimates (6.2.1b), (6.2.2b), (6.2.3), and (6.2.4), we obtain the desired bound (6.2.15). ˘X ˘ and use the comTo prove (6.2.16), we commute the evolution equation (6.2.14) with X mutator estimate (6.1.1b) with f = µ, and the already proven estimates (6.2.1b), (6.2.2b), and (6.2.3). It remains for us to prove the estimates |L /X˘ X˘ g/−1 | . 1 and |L /X˘ X˘ d/xi | . 1 that we used ˘ Xx ˘ i , we first note that Lemma 2.11 in proving (6.2.15). To handle the terms L /X˘ X˘ d/xi = d/X

J. Speck

71

˘ i = X ˘ i = f(γ). Thus, from the L∞ estimates of Prop. 5.6 and the already yields Xx ˘ Xx ˘ i | . |Z ≤2;1 γ| . 1 as desired. To handle the proven estimate (6.2.2b), we obtain |d/X −1 terms L /X˘ X˘ g/ , we rely on the basic identity L /X˘ g/−1 = −(L /X˘ g/)## , which was proved in [28, Lemma 2.9]. From this identity, Lemma 5.3, and the L∞ estimates of Prop. 5.6, we deduce that |L /X˘ X˘ g/−1 | . |L /X˘ X˘ g/| + 1. Moreover, Lemma 2.11 yields that g/ = f(γ, d/x1 , d /x2 ). Thus, from the L∞ estimates of Prop. 5.6, the already proven estimates (6.2.1b) and (6.2.3), ˘ Xx ˘ i | . 1 proved above, we conclude the desired bound |L and the bound |d/X /X˘ X˘ g/−1 | . 1. Proof of (6.2.5)-(6.2.6): Lemma 2.11 yields that GLL = f(γ). Thus, using the L∞ estimates of Prop. 5.6, we deduce |L(GLL )| . ε. The bound (6.2.5) now follows from this estimate and the mean value theorem. The proof of (6.2.6) is similar and is based on the relation ˘ = f(γ)XΨ; ˘ we omit the details. GLL XΨ Proof of (6.2.7): From equation (2.10.1), Lemma 2.11, and the L∞ estimates of Prop. 5.6, 1 ˘ u, ϑ) + O(ε). Next, from the identities we deduce that Lµ(t, u, ϑ) = GLL (t, u, ϑ)XΨ(t, 2 L0 = L0(F lat) = 1 and Li = Li(F lat) + Li(Small) and the L∞ estimates of Prop. 5.6, we deduce that GLL := Gαβ Lα Lβ = Gαβ (Ψ = 0)Lα Lβ + O(ε) = Gαβ (Ψ = 0)Lα(F lat) Lβ(F lat) + O(ε) := GL(F lat) L(F lat) (Ψ = 0) + O(ε). Combining these estimates with the simple bound ˘ L∞ (Σu ) . 1 (see (5.7.3a)), we conclude the desired bound (6.2.7). kXΨk t 7. Sharp Estimates for µ In this section, we derive sharp estimates for µ, its derivatives, and various time integrals, 1 many of which involve the singular factor . These estimates play a fundamental role in our µ energy estimates because our energies contain µ weights and because in our energy identities, 1 we will encounter error integrals that involve the derivatives of µ and/or factors of . The µ main results of this section are Props. 7.2 and Prop. 7.3. Remark 5.1 especially applies in this section. 7.1. Sharp sup-norm and pointwise estimates for the inverse foliation density. We define the following quantities in order to facilitate our analysis of µ. Definition 7.1 (Auxiliary quantities used to analyze µ). We define the following quantities, where 0 ≤ s ≤ t: Z s0 =t M (s, u, ϑ; t) := {Lµ(t, u, ϑ) − Lµ(s0 , u, ϑ)} ds0 , (7.1.1a) s0 =s

˚ µ(u, ϑ) := µ(s = 0, u, ϑ).

(7.1.1b)

As we outlined in Subsubsect. 1.7.6, our high-order energies are allowed to blow up as the shock forms. Specifically, the best estimates that we are able to derive allow for the possibility that the high-order energies blow up like negative powers of the quantity µ? , which we now define; see Prop. 9.7 for the detailed statement.

Shock formation for quasilinear wave systems featuring multiple speeds 72 Definition 7.2 (Definition of µ? ). µ? (t, u) := min{1, min µ}. u Σt

(7.1.2)

The following simple estimates play a role in our ensuing analysis. Lemma 7.1. [16, Lemma 11.1; First estimates for the auxiliary quantities] The following estimates hold for (t, u, ϑ) ∈ [0, T(Boot) ) × [0, U0 ] × T and 0 ≤ s ≤ t: ˚ µ(u, ϑ) = 1 + O(ε), ˚ µ(u, ϑ) = 1 + M (0, u, ϑ; t) + O(ε).

(7.1.3) (7.1.4)

To derive some of the most important estimates, we must distinguish regions in which µ is appreciably shrinking from regions in which it is not. We define the relevant regions in the next definition. Definition 7.3 (Regions of distinct µ behavior). For each t ∈ [0, T(Boot) ), s ∈ [0, t], and u ∈ [0, U0 ], we partition [0, u] × T = (+)Vtu ∪ (−)Vtu , Σus

=

(+)

Σus;t

∪

(−)

Σus;t ,

(7.1.5a) (7.1.5b)

where Lµ(t, u0 , ϑ) ≥0 , := (u , ϑ) ∈ [0, u] × T | ˚ µ(u0 , ϑ) − M (0, u0 , ϑ; t) Lµ(t, u0 , ϑ) (−) u 0 0 is the data-dependent constant from Def. 3.4.

(7.1.8)

J. Speck Upper bound for

73

˘ + [Xµ] . µ

[Xµ]

˘ +

µ

L∞ (Σu s)

C ≤p . T(Boot) − s

(7.1.9)

Sharp spatially uniform estimates. Consider a time interval s ∈ [0, t] and define the (t, u-dependent) constant κ by κ :=

[Lµ]− (t, u0 , ϑ) . µ(u0 , ϑ) − M (0, u0 , ϑ; t) (u0 ,ϑ)∈[0,u]×T ˚

(7.1.10)

sup

Note that κ ≥ 0 in view of the estimate (7.1.4). Then µ? (s, u) = {1 + O(ε)} {1 − κs} , ( √ 1 + O(ε1/2 ) κ, if κ ≥ ε, k[Lµ]− kL∞ (Σus ) = √ O(ε1/2 ), if κ ≤ ε.

(7.1.11a) (7.1.11b)

We also have κ ≤ {1 + O(ε)} ˚ δ∗ .

(7.1.12a)

κ = {1 + O(ε)} ˚ δ∗ .

(7.1.12b)

Moreover, when u = 1, we have

Sharp estimates when (u0 , ϑ) ∈ (+)Vtu . We recall that the set If 0 ≤ s1 ≤ s2 ≤ t, then the following estimate holds: sup (+)

Vtu is defined in (7.1.6a).

µ(s2 , u0 , ϑ) ≤ C. µ(s1 , u0 , ϑ)

(u0 ,ϑ)∈(+) Vtu

In addition, if s ∈ [0, t] and

(+)

(7.1.13)

Σus;t is as defined in (7.1.6c), then we have inf µ ≥ 1 − Cε.

(+) Σu s;t

Moreover, if s ∈ [0, t], then we have

[Lµ]−

µ

≤ Cε.

(7.1.14)

(7.1.15)

L∞ ((+) Σu s;t )

Sharp estimates when (u0 , ϑ) ∈ (−)Vtu . We recall that (−)Vtu is the set defined in (7.1.6b). Let κ > 0 be as in (7.1.10) and consider a time interval s ∈ [0, t]. Then the following estimate holds: µ(s2 , u0 , ϑ) sup ≤ 1 + Cε. (7.1.16) 0 0≤s1 ≤s2 ≤t µ(s1 , u , ϑ) (u0 ,ϑ)∈(−) Vtu

Shock formation for quasilinear wave systems featuring multiple speeds 74 Furthermore, if s ∈ [0, t] and holds:

(−)

Σus;t is as defined in (7.1.6d), then the following estimate

k[Lµ]+ kL∞ ((−)Σus;t ) ≤ Cε. Finally, there exists a constant C > 0 such that if 0 ≤ s ≤ t, then ( √ 1 + Cε1/2 κ, if κ ≥ ε, k[Lµ]− kL∞ ((−)Σus;t ) ≤ √ Cε1/2 , if κ ≤ ε.

(7.1.17)

(7.1.18)

Approximate time-monotonicity of µ−1 ? (s, u). There exists a constant C > 0 such that if 0 ≤ s1 ≤ s2 ≤ t, then −1 µ−1 ? (s1 , u) ≤ (1 + Cε)µ? (s2 , u).

(7.1.19)

Proof sketch. Thanks to the estimates of Props. 5.6 and 6.2, the proof of [16, Proposition 11.2] goes through verbatim. We note that some of the estimates of Prop. 6.2 are different than the analogous estimates proved in [17], the difference being that we do not gain a smallness factor of ε on the RHS of most of the estimates of Prop. 6.2. However, this difference has no substantial effect on the analysis since Prop. 6.2 is needed only in proving the estimate (7.1.9), which does not involve the smallness factor ε. 7.2. Sharp time-integral estimates involving µ. The following result, Prop. 7.3, plays a fundamental role in the Gronwall-type argument that we use in proving Prop. 9.7, which yields our main a priori energy estimates. Roughly, we use the estimates of Prop. 7.3 to control the integrating factor terms that we encounter in the Gronwall estimates. Proposition 7.3. [28, Proposition 10.2; Fundamental estimates for time integrals involving µ−1 ] Let µ? (t, u) be as defined in (7.1.2). Let B>1 be a real number. Then the following estimates hold for (t, u) ∈ [0, T(Boot) ] × [0, U0 ]. Estimates relevant for borderline top-order spacetime integrals. There exists a √ constant C > 0 such that if B ε ≤ 1, then √ Z t k[Lµ]− kL∞ (Σus ) 1 + C ε 1−B ds ≤ µ (t, u). (7.2.1) µB B−1 ? s=0 ? (s, u) Estimates relevant for borderline top-order hypersurface integrals. There exists √ a constant C > 0 such that if B ε ≤ 1, then √ Z t 1 1 + C ε 1−B kLµkL∞ ((−)Σut;t ) ds ≤ µ (t, u). (7.2.2) B B−1 ? s=0 µ? (s, u) Estimates relevant for less dangerous top-order spacetime integrals. There ex√ ists a constant C > 0 such that if B ε ≤ 1, then Z t 1 1 B ds ≤ C 2 + µ1−B (t, u). (7.2.3) ? B (s, u) µ B − 1 s=0 ?

J. Speck

75

degeneracy. There exists a constant Estimates for integrals that lead to only ln µ−1 ? C > 0 such that Z t k[Lµ]− kL∞ (Σus ) √ √ ds ≤ (1 + C ε) ln µ−1 (7.2.4) ? (t, u) + C ε. µ? (s, u) s=0 In addition, there exists a constant C > 0 such that Z t 1 ds ≤ C ln µ−1 ? (t, u) + 1 . s=0 µ? (s, u)

(7.2.5)

Estimates for integrals that break the µ−1 degeneracy. There exists a constant C > ? 0 such that Z t 1 ds ≤ C. (7.2.6) 9/10 (s, u) s=0 µ? Proof. Thanks to the estimates of Prop. 7.2, the proof of [17, Proposition 11.3] goes through verbatim. 8. Pointwise estimates In this section, we use some estimates that we established in prior sections to derive pointwise estimates for the error terms that we encounter in our energy estimates. Remark 5.1 especially applies in this section. 8.1. Harmless error terms. Most error terms that we encounter are harmless in the sense that they remain negligible all the way up to the shock. We now precisely define what we mean by “harmless.” Definition 8.1 (Harmless terms). A Harmless≤N term is any term such that under the data-size and bootstrap assumptions of Sects. 3.5-3.6 and the smallness assumptions of Sect. 3.8, the following bound holds on MT(Boot) ,U0 (see Subsect. 3.2 regarding the vectorfield operator notation): Harmless≤N . Z∗≤N +1;1 Ψ + Z∗≤N ;1 γ + P∗[1,N ] γ . (8.1.1) A Harmless≤N (Slow) term is any term such that under the data-size and bootstrap assumptions of Sects. 3.5-3.6 and the smallness assumptions of Sect. 3.8, the following bound holds on MT(Boot) ,U0 : ≤N ~ ≤N (8.1.2) Harmless(Slow) . P W . Remark 8.1. Our definition of Harmless≤N terms is the same as in [28] while our definition of Harmless≤N (Slow) terms accounts for the harmless error terms corresponding to the slow wave ~ variable W .

Shock formation for quasilinear wave systems featuring multiple speeds 76 8.2. Identification of the key difficult error terms in the commuted equations. As we mentioned, most error terms that arise upon commuting the wave equations are negligible. In the next proposition, we identify those error terms that are not. Proposition 8.1 (Identification of the key difficult error term factors). Recall that ρ is the scalar function from Lemma 2.7. For 1 ≤ N ≤ 18, we have the following estimates: µg (Y N −1 LΨ) = (d/# Ψ) · (µd/Y N −1 trg/ χ) + Harmless≤N + Harmless≤N (Slow) ,

(8.2.1a)

≤N N ˘ µg (Y N Ψ) = (XΨ)Y trg/ χ + ρ(d/# Ψ) · (µd/Y N −1 trg/ χ) + Harmless≤N + Harmless(Slow) . (8.2.1b)

Furthermore, if 2 ≤ N ≤ 18 and P N is any N th order Pu -tangent operator except for Y N −1 L or Y N , then µg (P N Ψ) = Harmless≤N + Harmless≤N (Slow) .

(8.2.1c)

Moreover, for 1 ≤ N ≤ 18, we have the following estimates: µ∂t P N w0 = µ(h−1 )ab ∂a P N wb + 2µ(h−1 )0a ∂a P N w0 + Harmless≤N + Harmless≤N (Slow) , (8.2.2a) µ∂t wi = µ∂i w0 + Harmless≤N + Harmless≤N (Slow) ,

(8.2.2b)

µ∂t w = µw0 + Harmless≤N + Harmless≤N (Slow) ,

(8.2.2c)

µ∂i wj = µ∂j wi + Harmless≤N + Harmless≤N (Slow) .

(8.2.2d)

Proof. See Subsect. 5.2 for some comments on the analysis. We first establish (8.2.2a). From Lemmas 2.11 and 2.13 and the assumptions on the inhomogeneous terms stated in (1.6.5), we see that the products of µ and the inhomogeneous terms on the second line of ~ XΨ, ˘ P Ψ)P Ψ+f(γ, W, ~ XΨ, ˘ P Ψ)Ψ+f(γ, W, ~ XΨ, ˘ P Ψ)W ~. RHS (1.6.11a) are of the form f(γ, W, Thus, from the L∞ estimates of Prop. 5.6, we find that the P N derivatives of these terms are ≤N +1;1 ≤N [1,N ] ~ + P∗ γ = Harmless≤N + Harmless≤N . To in magnitude . Z∗ Ψ + P W (Slow) complete the proof of (8.2.2a), it remains for us to bound the commutator terms [µ∂t , P N ]w0 , [(h−1 )ab µ∂a , P N ]wb , and [(h−1 )0a µ∂a , P N ]w0 . We show how to bound the last one; the first two can be bounded similarly. From Lemmas 2.11 and 2.13 and the fact that (h−1 )αβ = ~ ), we see that we must show that [f(γ, W ~ )P, P N ]w0 + [f(γ, W ~ )X, ˘ P N ]w0 = (h−1 )αβ (Ψ, W Harmless≤N + Harmless≤N (Slow) . The desired bounds follow from commutator estimates of Lemmas 5.4 and 5.5 with f = w0 , the algebraic identity provided by Lemma 2.14 (which ˘W ~ with P W ~ up to error terms), and the L∞ estimates of Prop. 5.6. allows us to replace X To derive (8.2.2b)-(8.2.2d), we use the same reasoning that we used in the previous paragraph; the analysis is even simpler since, in view of the absence of inhomogeneous semilinear terms on RHSs (1.6.11b)-(1.6.11d), we encounter only commutator error terms. We now prove the estimates (8.2.1a)-(8.2.1c). Theses estimates were derived in the proof of [28, Proposition 11.2] with the help of the L∞ estimates of Prop. 5.6, except that the derivatives of µ× the inhomogeneous semilinear terms on RHS (1.6.2a) were not treated there (because these terms were absent from [28]). To handle µ× these semilinear terms, we

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first use Lemma 2.11, Lemma 2.13, and the assumptions (1.6.5) to deduce that the products ~ XΨ, ˘ P Ψ)P Ψ + f(γ, W, ~ XΨ, ˘ P Ψ)Ψ + under consideration are of the schematic form f(γ, W, ~ XΨ, ˘ P Ψ)W ~ . Thus, for the same reasons given two paragraphs above, the P N derivaf(γ, W, tives of these quantities are Harmless≤N + Harmless≤N (Slow) as desired.

8.3. Pointwise estimates for the most difficult product. The first product Y N trg/ χ on RHS (8.2.1b) is the most difficult one to control in the energy estimates. In the next proposition, we derive estimates for it. We also derive pointwise estimates for the Npointwise easier error term µY trg/ χ . Compared to previous works, the estimates of the proposition ~ on trg/ χ, that is, involve new terms stemming from the influence of the slow wave variable W on the null mean curvature of the characteristics. N ˘ Proposition 8.2 (The key pointwise estimate for (XΨ)Y trg/ χ). Let

(Y N )

X := µY trg/ χ + Y N

N

1 1 # ˘ −GLL XΨ − µtrg/ G / LΨ − µGLL LΨ + µG /L · d/Ψ . 2 2

(8.3.1)

We have the following pointwise estimate:

[Lµ] ˘ −

(XΨ)Y N trg/ χ ≤ 2

µ

L∞ (Σu t)

+ 4 (1 + Cε)

˘ N XY Ψ

(8.3.2)

k[Lµ]− kL∞ (Σut ) Z µ? (t, u)

t

t0 =0

k[Lµ]− kL∞ (Σu0 ) N 0 t ˘ XY Ψ (t , u, ϑ) dt0 0 µ? (t , u)

+ Error, where 1 (Y N ) |Error| . X (0, u, ϑ) + Z∗≤N +1;1 Ψ (8.3.3) µ? (t, u) ≤N ;1 ≤N [1,N ] 1 1 1 Z∗ P γ + P∗ γ + Ψ + µ? (t, u) µ? (t, u) µ? (t, u) Z t 1 1 ˘ N 0 +ε XP Ψ (t , u, ϑ) dt0 µ? (t, u) t0 =0 µ? (t0 , u) Z t ≤N +1;1 0 1 Z∗ Ψ (t , u, ϑ) dt0 + µ? (t, u) t0 =0 Z t ≤N ;1 ≤N [1,N ] 0 1 1 Z∗ + Ψ + P γ + P∗ γ (t , u, ϑ) dt0 µ? (t, u) t0 =0 µ? (t0 , u) Z t 1 ≤N ~ 0 + P W (t , u, ϑ) dt0 . µ? (t, u) t0 =0

Shock formation for quasilinear wave systems featuring multiple speeds 78 Furthermore, we have the following less precise pointwise estimate: N µY trg/ χ (t, u, ϑ) (8.3.4) N ˘ N . (Y )X (0, u, ϑ) + µ P N +1 Ψ (t, u, ϑ) + XP Ψ (t, u, ϑ) + Z∗≤N ;1 Ψ (t, u, ϑ) + P ≤N γ (t, u, ϑ) + P∗[1,N ] γ (t, u, ϑ) Z t Z t ≤N +1;1 0 1 ˘ N 0 0 Z∗ + Ψ (t , u, ϑ) dt0 XP Ψ (t , u, ϑ) dt + 0 t0 =0 µ? (t , u) t0 =0 Z t 1 Z∗≤N ;1 Ψ + P ≤N γ + P∗[1,N ] γ (t0 , u, ϑ) dt0 + 0 t0 =0 µ? (t , u) Z t ≤N ~ 0 + P W (t , u, ϑ) dt0 . t0 =0

Proof sketch. The estimate (8.3.2) was essentially obtained in [28, Proposition 11.10] with the help of the L∞ estimates of Prop. 5.6, the estimate (6.2.5), and the estimates of Prop. 7.2. We note that on RHS (8.3.3), we have corrected a typo that appeared in [28]. Specifically, Z t 1 1 ˘ N ˘ N 0 the factor XP Ψ in the term ε XP Ψ (t , u, ϑ) dt0 on RHS (8.3.3) 0 , u) µ (t, u) µ (t 0 ? ? t =0 was incorrectly listed as Z∗≤N +1;1 Ψ in [28, Equation (11.33)]. The only new term appearing on RHS (8.3.2) compared to [28, Proposition 11.10] is the last one on RHS (8.3.3) (which ≤N ~ involves the time integral of P W ), whose origin we now explain. To do this, we must explain some features of the proof of (8.3.2), which relies on the “modified” version of N trg/ χ described in Subsubsect. 1.7.6, namely the quantity (Y )X defined in (8.3.1). As we N explained in the discussion below equation (1.7.40), we are forced to work with (Y )X in order to avoid losing a derivative at the top-order. Specifically, to prove (8.3.2), one first N derives a transport equation for (Y )X (see the proof of [28, Lemma 11.9] for more details) N of the form L(Y )X = · · · , where · · · contains, among other terms, µ × RHS (1.6.2a); the terms µ × RHS (1.6.2a) are the new ones compared to the terms found in [28, Proposition ~ XΨ, ˘ P Ψ)P Ψ + 11.10]. Using Lemmas 2.11 and 2.13, we see that µ × RHS (1.6.2a) = f(γ, W, ~ XΨ, ˘ P Ψ)W ~ + f(γ, W, ~ XΨ, ˘ P Ψ)Ψ. Therefore, with the help of the L∞ estimates of f(γ, W, Prop. 5.6, can pointwise bound these new terms in magnitude by . Z∗≤N +1;1 Ψ + we ≤N ~ [1,N ] P W + P∗ γ . Revisiting the proof of [28, Lemma 11.9], which is based on integrating the evolution equation L(Y )X = · · · in time, we obtain, using the Z t N for the new terms in · · · , the following estimate: µY trg/ χ ≤ C N

above pointwise bounds ≤N ~ 0 P W (t , u, ϑ) dt0 +

t0 =0

· · · , where · · · now denotes terms of the same type that appeared in the proof of [28, Lemma 11.9]. It follows that Z t 1 ˘ ˘ ≤N ~ 0 N (8.3.5) (XΨ)Y trg/ χ ≤ C XΨ (t, u, ϑ) P W (t , u, ϑ) dt0 + · · · µ(t, u, ϑ) t0 =0 Z t 1 ≤N ~ 0 P W ≤C (t , u, ϑ) dt0 + · · · , µ? (t, u) t0 =0

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79

where · · · again denotes terms that appear in the proof of [28, Lemma 11.9] (and thus on RHS (8.3.2) as well) and to obtain the last inequality in (8.3.5), we used the simple bound ˘ L∞ (Σu ) . 1 (that is, (5.7.3a)). This explains the origin of the last term on RHS (8.3.3) kXΨk t and completes our proof sketch of (8.3.2). Similarly, the estimate (8.3.4) was essentially obtained in [28, Proposition 11.10] using ideas similar to but simpler than the ones used in the proof of (8.3.2). The new terms mentioned in the previous paragraph also make a contribution to RHS (8.3.4) for essentially the same reason that they appeared on RHS (8.3.5). Specifically, they lead to the last term 1 on RHS (8.3.4). Note that RHS (8.3.4) is less singular with respect to factors of compared µ to RHS (8.3.2). The reason is that LHS (8.3.4) has an extra factor of µ in it compared to LHS (8.3.2). 8.4. Pointwise estimates for the remaining terms in the energy estimates. We now derive pointwise estimates for the energy estimates error integrands (T )P(i) [f ] from RHS (4.3.1) and the error integrand {1 + γf(γ)} W[V~ ] from (4.3.4). Lemma 8.3 (Pointwise bounds for the remaining error terms in the energy estimates). Consider the error terms (T )P(1) [f ], · · · , (T )P(5) [f ] defined in (4.3.2a)-(4.3.2e). Let ς > 0 be a real number. Then the following pointwise estimate holds without any absolute value taken on the left, where the implicit constants are independent of ς: 5 X

(T )

˘ )2 + µ|d/f |2 + ς˚ P(i) [f ] . (1 + ς −1 )(Lf )2 + (1 + ς −1 )(Xf δ∗ |d/f |2

(8.4.1)

i=1

1 +p µ|d/f |2 . T(Boot) − t In addition, we have the following pointwise estimate for the error integrand {1 + γf(γ)} W[V~ ] on RHS (4.3.4), where W[V~ ] is defined by (4.3.5): (8.4.2) {1 + γf(γ)} W[V~ ] . |V~ |2 . Remark 8.2. In deriving energy estimates, we will rely on the estimate (8.4.1) with P N Ψ ~ in the role of V~ . in the role of f and the estimate (8.4.2) with P N W Proof. The estimate (8.4.1) was proved as [28, Lemma 11.11] on the basis of the L∞ estimates of Prop. 5.6 and the sharp estimates of Prop. 7.2. ~) = To prove (8.4.2), we first use Lemmas 2.11 and 2.13 to deduce that µ∂κ (h−1 )αβ (Ψ, W ˘ + f(γ)P W ~ + f(γ)X ˘W ~ . From this schematic identity and the L∞ estif(γ)P Ψ + f(γ)XΨ ~ ) . 1, µ∂κ (h−1 )αβ (Ψ, W ~ ) . 1, mates of Prop. 5.6, we obtain the bounds (h−1 )αβ (Ψ, W |1 + γf(γ)| . 1, and µ . 1, from which the desired estimate (8.4.2) easily follows.

Shock formation for quasilinear wave systems featuring multiple speeds 80 9. Energy estimates and improvements of the fundamental bootstrap assumptions In this section, we derive the main estimates of this article: a priori L2 estimates for the solution up to top-order. As a simple corollary, we will also derive strict improvements of ~ ). Remark 5.1 especially applies in the fundamental L∞ bootstrap assumptions (BAΨ − W this section. 9.1. Definitions of the fundamental square-integral controlling quantities. In this subsection, we define the quantities that we use to control the solution in L2 up to top-order. Definition 9.1 (The main coercive quantities used for controlling the solution and its derivatives in L2 ). In terms of the energies and null fluxes of Defs. 4.1 and 4.2, we define o n ~ ~ (9.1.1a) QN (t, u) := max sup E(F ast) [P I Ψ](t0 , u0 ) + F(F ast) [P I Ψ](t0 , u0 ) , ~ |I|=N (t0 ,u0 )∈[0,t]×[0,u]

Q[1,N ] (t, u) := max QM (t, u),

(9.1.1b)

1≤M ≤N

WN (t, u) := max

sup

~ |I|=N (t0 ,u0 )∈[0,t]×[0,u]

n o ~~ ~~ E(Slow) [P I W ](t0 , u0 ) + F(Slow) [P I W ](t0 , u0 ) , (9.1.1c)

W[1,N ] (t, u) := max WM (t, u).

(9.1.1d)

1≤M ≤N

We use the following coercive spacetime integrals to control non−µ-weighted error integrals Z involving geometric torus derivatives. These integrals are generated by the term 1 [Lµ]− |d/f |2 on the RHS of the fast wave energy identity (4.3.1). − 2 Mt,u Definition 9.2 (Key coercive spacetime integrals). We associate the following integrals to Ψ, where [Lµ]− = |Lµ| when Lµ < 0 and [Lµ]− = 0 when Lµ ≥ 0: Z 1 K[Ψ](t, u) := [Lµ]− |d/Ψ|2 d$, (9.1.2a) 2 Mt,u ~

KN (t, u) := max K[P I Ψ](t, u), ~ |I|=N

K[1,N ] (t, u) := max KM (t, u). 1≤M ≤N

(9.1.2b) (9.1.2c)

9.2. The coerciveness of the fundamental L2 -controlling quantities. 9.2.1. Preliminary lemmas. We start with a lemma that provides identities for the derivatives of various integrals over the tori `t,u . Lemma 9.1. [28, Lemma 3.6; Identities connected to integration by parts] The following identities hold for scalar-valued functions f : Z Z ∂ f dλg/ = Lf + trg/ χf dλg/ , (9.2.1a) ∂t `t,u `t,u Z Z ∂ 1 (X) ˘ ˘ f dλg/ = Xf + trg/ π /f dλg/ . (9.2.1b) ∂u `t,u 2 `t,u

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81

We now provide a lemma in which we compare the strength of the area forms d$ at different times. Lemma 9.2. [28, Lemma 12.3; Comparison of L2 norms at different times] Let p = p(u0 , ϑ) be a non-negative function of (u0 , ϑ) ∈ [0, u] × T that does not depend on t. Then for s, t ∈ [0, T(Boot) ) and u ∈ [0, U0 ], we have the following estimate: Z Z Z p d$ ≤ p d$ ≤ (1 + Cε) p d$. (1 − Cε) (9.2.2) Σu t

Σu s

Σu s

We now provide a simple variant of Minkowski’s inequality for integrals. Lemma 9.3. [28, Lemma 13.2; Estimate for the norm k · kL2 (Σut ) of time-integrated functions] Let f be a scalar function on MT(Boot) ,U0 and let Z t f (t0 , u, ϑ) dt0 . (9.2.3) F (t, u, ϑ) := t0 =0

We have the following estimate Z kF k

L2 (Σu t)

t

≤ (1 + Cε) t0 =0

kf kL2 (Σut0 ) dt0 .

(9.2.4)

9.2.2. The coerciveness of the fundamental L2 -controlling quantities. We now provide the main lemma of Subsect. 9.2. Lemma 9.4 (The coerciveness of the fundamental L2 -controlling quantities). Let 1 ≤ M ≤ N ≤ 18, and let P M be an M th -order Pu -tangent vectorfield operator. We have the following lower bounds for (t, u) ∈ [0, T(Boot) ) × [0, U0 ]:

2 n 1 √

1 ˘ M Ψ

µLP M Ψ 2 2 u ,

√µd/P M Ψ 2 2 u , Q[1,N ] (t, u) ≥ max XP ,

L (Σt ) L (Σt ) 2 2 L2 (Σu t) (9.2.5) o

LP M Ψ 2 2 t , √µd/P M Ψ 2 2 t . L (P ) L (P ) u

u

Moreover, if 1 ≤ M ≤ N ≤ 18, then the following bounds hold: 1/2

+ CQ1 (t, u), kΨkL2 (Σut ) ≤ C˚

M

1/2

P Ψ 2 u , P M Ψ 2 ≤ C˚ + CQ[1,N ] (t, u), L (Σt ) L (`t,u )

˘

˘ 1/2 XΨ ≤ C XΨ + CQ1 (t, u).

2 u

2 u + C˚ L (Σt )

(9.2.6a) (9.2.6b) (9.2.6c)

L (Σ0 )

In addition, with 1{µ≤1/4} denoting the characteristic function of the spacetime subset {(t, u, ϑ) ∈ [0, ∞) × [0, 1] × T | µ(t, u, ϑ) ≤ 1/4}, then for 1 ≤ M ≤ N ≤ 18, we have the following bound: Z 2 1˚ K[1,N ] (t, u) ≥ δ∗ 1{µ≤1/4} d/P M Ψ d$. (9.2.7) 8 Mt,u In addition, for 1 ≤ M ≤ N ≤ 18, we have the following bounds:

2

1 1

√

M ~ 2 M ~ W[1,N ] (t, u) ≥ µP W + P W . C C L2 (Σu L2 (Put ) t)

(9.2.8)

Shock formation for quasilinear wave systems featuring multiple speeds 82 Finally, for 0 ≤ M ≤ N − 1 ≤ 17, we have the following bounds:

M~

M~ 1/2 P W , P W ≤˚ + W[1,N ] (t, u).

2 u

2 L (Σt )

(9.2.9)

L (`t,u )

Proof sketch. All bounds except for (9.2.8)-(9.2.9) follow from the proofs of [16, Lemma 12.4] and [16, Lemma 12.5]. The estimate (9.2.8) is a simple consequence of Lemma 4.1 and definition (9.1.1d).

M~ To prove (9.2.9), we first note that it suffices to obtain the desired estimate for P W . L2 (`t,u )

2

~ with reThe reason is that we can integrate the corresponding estimate for P M W

2 L (`t,u )

2

~ spect to u to obtain the desired bound for P M W

2 u . To obtain the desired bound L (Σt ) 2

~ , Young’s inequality, and ~ , we first use (9.2.1a) with f = P M W (9.2.9) for P M W

L2 (`t,u )

the estimate (5.7.8a) to obtain Z Z ∂ M ~ 2 M ~ 2 M ~ 2 L P W + trg/ χ P W dλg/ P W dλg/ = ∂t `t,u `t,u Z Z 2 M ~ 2 M ~ ≤ LP W dλg/ + C P W dλg/ . `t,u

(9.2.10)

`t,u

Integrating (9.2.10) with respect to time starting from time 0, we obtain

M ~ 2

P W

L2 (`t,u )

M ~ 2 ≤ P W

L2 (`

0,u )

2

M ~ + LP W

L2 (Put )

Z

t

+C t0 =0

M ~ 2

P W

L2 (`t0 ,u )

dt0 . (9.2.11)

From the small-data assumption (3.3.14), we see that the first term on RHS (9.2.11) is 2

~ is . WM +1 (t, u) . .˚ 2 , while from (9.2.8), we see that the second term LP M W

L2 (Put )

W[1,N ] (t, u). Using these bounds, we apply Gronwall’s inequality to (9.2.11), which yields

2

~ the desired estimate P M W . W[1,N ] (t, u) + ˚ 2 .

L2 (`t,u )

9.2.3. The initial smallness of the fundamental L2 -controlling quantities. The next lemma shows that the fundamental L2 -controlling quantities of Def. 9.1 are initially small. Lemma 9.5 (The fundamental controlling quantities are initially small). Assume that 1 ≤ N ≤ 18. Under the data-size assumptions of Subsect. 3.5, the following estimates hold for (t, u) ∈ [0, 2˚ δ−1 ∗ ] × [0, U0 ]: Q[1,N ] (0, u), Q[1,N ] (t, 0) . ˚ 2 ,

(9.2.12a)

W[1,N ] (0, u), W[1,N ] (t, 0) . ˚ 2 .

(9.2.12b)

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83 ˚−1

Proof. We first note that by (3.4.4a), we have µ ≈ 1 along Σ10 and along P02δ∗ . Using these estimates, (4.1.1a), and Def. 9.1, we see that

2 Q[1,18] (0, 1) . Z∗[1,19];≤1 Ψ L2 (Σ1 ) , 0

[1,19] 2

Ψ L2 (P 2˚δ−1 Q[1,18] (2˚ δ−1 ∗ ) , ∗ , 0) . P 0

[1,18] ~ 2 , W[1,18] (0, 1) . P W L2 (Σ10 )

[1,18] ~ 2 W[1,18] (2˚ δ−1 , 0) . P W .

2 2˚δ−1 ∗ ∗ L (P0

(9.2.13) (9.2.14)

)

The estimates (9.2.12a)-(9.2.12b) now follow from (9.2.13)-(9.2.14) and the initial data assumptions (3.3.2), (3.3.3), (3.3.8), and (3.3.9).

9.3. The main a priori energy estimates. In this subsection, we state our main a priori energy estimates. The main step in their proof is to derive L2 estimates for the error terms in the commuted equations; we carry out this technical analysis in later subsections.

9.3.1. The system of integral inequalities verified by the energies. We start with a proposition in which we provide the system of integral inequalities verified by the energies. Its proof is located in Subsect. 9.8.

Proposition 9.6 (Integral inequalities for the fundamental L2 -controlling quantities). Assume that 1 ≤ N ≤ 18 and ς > 0. There exists a constant C > 0, independent of

Shock formation for quasilinear wave systems featuring multiple speeds 84 ς, such that following estimates hold for (t, u) ∈ [0, T(Boot) ) × [0, U0 ]: max Q[1,N ] (t, u), K[1,N ] (t, u), W[1,N ] (t, u) −1

)˚ 2 µ−3/2 (t, u) ?

(9.3.1a)

≤ C(1 + ς Z t k[Lµ] k ∞ u − L (Σ 0 ) t + 6 Q[1,N ] (t0 , u) dt0 0 , u) µ (t 0 ? t =0 Z t k[Lµ] k ∞ u Z t0 k[Lµ]− kL∞ (Σus ) 1/2 − L (Σ 0 ) 1/2 0 t + 8.1 Q (t , u) Q[1,N ] (s, u) ds dt0 [1,N ] 0 , u) µ (t µ (s, u) 0 ? ? s=0 t =0 Z t 1 1 1/2 1/2 + 2 1/2 Q[1,N ] (t, u) kLµkL∞ ((−)Σut;t ) Q[1,N ] (t0 , u) dt0 1/2 µ? (t, u) t0 =0 µ? (t0 , u) Z t Z t0 1 1 1/2 1/2 + Cε Q[1,N ] (t0 , u) Q[1,N ] (s, u) ds dt0 0 µ (t , u) µ (s, u) 0 t =0 ? s=0 ? Z t 1 Q[1,N ] (t0 , u) dt0 + Cε 0 t0 =0 µ? (t , u) Z t 1 1 1/2 1/2 + Cε 1/2 Q[1,N ] (t, u) Q[1,N ] (t0 , u) dt0 1/2 µ? (t, u) t0 =0 µ? (t0 , u) Z t 1 1/2 1/2 + CQ[1,N ] (t, u) Q[1,N ] (t0 , u) dt0 1/2 0 0 t =0 µ? (t , u) Z t 1 p +C Q (t0 , u) dt0 0 [1,N ] 0 T − t t =0 (Boot) Z t 1 −1 Q[1,N ] (t0 , u) dt0 + C(1 + ς ) 1/2 0 0 t =0 µ? (t , u) Z t Z t0 1 1 1/2 1/2 0 +C Q (t , u) Q[1,N ] (s, u) ds dt0 [1,N ] 1/2 0 t0 =0 µ? (t , u) s=0 µ? (s, u) Z t Z s Z t0 1 1 1 1/2 1/2 0 +C Q[1,N ] (t , u) Q[1,N ] (s0 , u) ds0 ds dt0 1/2 0 t0 =0 µ? (t , u) s=0 µ? (s, u) s0 =0 µ? (s0 , u) Z u + C(1 + ς −1 ) Q[1,N ] (t, u0 ) du0 u0 =0

+ CεQ[1,N ] (t, u) + CςQ[1,N ] (t, u) + CςK[1,N ] (t, u) Z t 1 +C Q[1,N −1] (t0 , u) dt0 5/2 0 t0 =0 µ? (t , u) Z t Z t0 1 1 1/2 1/2 0 +C Q[1,N ] (t , u) W[1,N ] (s, u) ds dt0 1/2 0 t0 =0 µ? (t , u) s=0 µ? (s, u) Z u + C(1 + ς −1 ) W[1,N ] (t, u0 ), du0 . u0 =0

J. Speck

85

Moreover, if 2 ≤ N ≤ 18, then we have the following estimates: max Q[1,N −1] (t, u), K[1,N −1] (t, u), W[1,N −1] (t, u) ≤ C˚ 2 Z +C

t

1

1/2 t0 =0 µ? (t0 , u) Z t

1/2 Q[1,N −1] (t0 , u)

Z

t0

1

1/2 s=0 µ? (s, u)

(9.3.1b)

1/2

Q[1,N ] (s, u) ds dt0

1 p Q[1,N −1] (t0 , u) dt0 0 0 T(Boot) − t t =0 Z t 1 −1 + C(1 + ς ) Q[1,N −1] (t0 , u) dt0 1/2 0 0 t =0 µ? (t , u) Z t 1 1/2 Q[1,N −1] (t0 , u) dt0 + C˚ 1/2 0 t0 =0 µ? (t , u) Z u −1 + C(1 + ς ) Q[1,N −1] (t, u0 ) du0 +C

u0 =0

+ CςK[1,N −1] (t, u), Z u −1 W[1,N −1] (t, u0 ) du0 . + C(1 + ς ) u0 =0

9.3.2. The main a priori energy estimates. We now provide the main a priori energy estimates. Proposition 9.7 (The main a priori energy estimates). There exists a constant C > 0 such that under the data-size and bootstrap assumptions of Sects. 3.5-3.6 and the smallness assumptions of Sect. 3.8, the following estimates hold for (t, u) ∈ [0, T(Boot) ) × [0, U0 ]: 1/2

1/2

1/2

+.9) µ−(M (t, u), Q[1,13+M ] (t, u) + K[1,13+M ] (t, u) + W[1,13+M ] (t, u) ≤ C˚ ?

1/2

1/2

1/2

Q[1,12] (t, u) + K[1,12] (t, u) + W[1,12] (t, u) ≤ C˚ .

(0 ≤ M ≤ 5), (9.3.2a) (9.3.2b)

Proof. Based on the inequalities of Prop. 9.6, the proof of [28, Proposition 14.1] applies with only very minor changes needed to account for the terms depending on WN , N = 1, 2, · · · , 18. In fact, if one views the quantities max Q (t, u), K (t, u), W (t, u) [1,N ] [1,N ] [1,N ] and max Q[1,N −1] (t, u), K[1,N −1] (t, u), W[1,N −1] (t, u) to be the unknowns in the system of inequalities (9.3.1a)-(9.3.1b), then the proof of [28, Proposition 14.1] goes through almost verbatim. For this reason, we omit the details, noting only that the sharp estimates of Prop. 7.3 are essential for handling the “boxed-constant-involving” terms from Prop. 9.6. 9.3.3. Improvement of the fundamental bootstrap assumptions. Using the energy estimates provided by Prop. 9.7, we can strictly improve the fundamental L∞ bootstrap assumptions ~ ). The main ingredient in this vein is the following simple Sobolev embedding (BAΨ − W result.

Shock formation for quasilinear wave systems featuring multiple speeds 86 Lemma 9.8. [28, Lemma 12.4; Sobolev embedding along `t,u ] The following estimate holds for scalar-valued functions f defined on `t,u for (t, u) ∈ [0, T(Boot) ) × [0, U0 ]:

kf kL∞ (`t,u ) ≤ C Y ≤1 f L2 (`t,u ) .

(9.3.3)

~ ). We now derive improvements of the bootstrap assumptions (BAΨ − W Corollary 9.9 (Improvement of the fundamental L∞ bootstrap assumptions). The ~ ) stated in Subsect. 3.6 hold with RHS (BAΨ − W ~) fundamental bootstrap assumptions (BAΨ − W replaced by C˚ . In particular, if ˚ is sufficiently small compared to ε, then we have obtained ~ ) on MT a strict improvement of the bootstrap assumptions (BAΨ − W . (Boot) ,U0 Proof. From Lemma 9.4, the a priori energy estimates stated in (9.3.2b), and the Sobolev embedding result (9.3.3), we deduce that

[1,11] 1/2

P Ψ L∞ (Σu ) . Q[1,12] (t, u) . ˚ , t

≤10 ~ P W

L∞ (Σu t)

1/2

. W[1,12] (t, u) . ˚ .

(9.3.4)

Moreover, a special case of the estimates in (9.3.4) is the bound kLΨkL∞ (Σut ) . ˚ . From this bound, the fundamental theorem ofRcalculus, and the small-data assumption (3.3.4), we t deduce that kΨkL∞ (Σut ) . kΨkL∞ (Σu ) + t0 =0 kLΨkL∞ (Σu0 ) dt0 . ˚ . This completes the proof 0 t of the corollary. Remark 9.1 (The main step in the article). In view of Cor. 9.9, we have justified the fundamental L∞ bootstrap assumptions until the time of first shock formation. This is the main step in the article. 9.4. Estimates for the most difficult top-order energy estimate error term. It remains for us to prove Prop. 9.6. The proof is located in Subsect. 9.8. To prove the proposition, we must bound all of the error integrals appearing in the energy identities (up to top-order) of Props. 4.2 and 4.3 in terms of the fundamental L2 -controlling quantities. The error integrals are generated by the inhomogeneous terms on the RHS of the equations of Prop. 8.1 as well as the integrands from Lemma 8.3 (see also Remark 8.2). As an important first step, in the next lemma, we bound the L2 norm of the most difficult N ˘ product that we encounter, namely the product (XΨ)Y trg/ χ on the RHS of the wave equaN tion (8.2.1b) verified by Y Ψ. The proof of the lemma is based on the pointwise estimates of Prop. 8.2, which in turn was based on the modified quantities described in Subsubsect. 1.7.6; we recall that the modified quantity (8.3.1) was needed in the proof of Prop. 8.2 in order to avoid the loss of a derivative. Lemma 9.10 (L2 bound for the most difficult product). Assume that 1 ≤ N ≤ 18. There exists a constant C > 0 such that the following L2 estimate holds for the difficult

J. Speck

87

N ˘ product (XΨ)Y trg/ χ from Prop. 8.2:

k[Lµ]− kL∞ (Σut ) 1/2

˘

Q[1,N ] (t, u) (9.4.1)

(XΨ)Y N trg/ χ 2 u ≤ 2 µ? (t, u) L (Σt ) k[Lµ]− kL∞ (Σut ) Z t k[Lµ]− kL∞ (Σus ) 1/2 Q[1,N ] (s, u) ds + 4.05 µ? (t, u) µ? (s, u) s=0 Z t 1 1 1/2 + Cε Q[1,N ] (s, u) ds µ? (t, u) s=0 µ? (s, u) Z t Z s0 1 1 1 1/2 +C (s, u) ds ds0 Q 1/2 0 µ? (t, u) s0 =0 µ? (s , u) s=0 µ? (s, u) [1,N ] Z t 1 1 1/2 +C Q (s, u) ds 1/2 µ? (t, u) s=0 µ? (s, u) [1,N ] 1 1 1/2 1/2 + C 1/2 Q[1,N ] (t, u) + C 3/2 Q[1,N −1] (t, u) µ? (t, u) µ? (t, u) Z t 1 1 1/2 +C W[1,N ] (s, u) ds 1/2 µ? (t, u) s=0 µ? (s, u) 1 ˚ + C 3/2 . µ? (t, u)

Furthermore, we have the following less precise estimate: Z t

N

1 1/2

µY trg/ χ 2 u . Q1/2 (t, u) + Q[1,N ] (s, u) ds [1,N ] L (Σt ) s=0 µ? (s, u) Z t 1 1/2 −1 ˚ + W (s, u) ds + ln µ (t, u) + 1 . ? [1,N ] 1/2 s=0 µ? (s, u)

(9.4.2)

Proof sketch. To prove (9.4.1), we start by taking the norm k · kL2 (Σut ) of both sides of inequality (8.3.2). The norms k · kL2 (Σut ) of all terms on RHS (8.3.2) were bounded in the proof of [28, Lemma 14.8] (the typo correction mentioned in the proof Z of Prop. 8.2 is important t 1 ≤N ~ 0 for the proof of [28, Lemma 14.8]) except for the last term P W (t , u, ϑ) ds µ? (t, u) t0 =0 on RHS (8.3.3). To handle this last term, we first use (9.2.4) to bound its norm k · kL2 (Σut ) by Z t

1

≤N ~ ≤C (9.4.3)

P W u ds. µ? (t, u) s=0 Σs Using (9.2.8)-(9.2.9), we deduce that 1 1 ˚ RHS (9.4.3) ≤ C +C µ? (t, u) µ? (t, u)

Z

t

1

1/2 s=0 µ? (s, u)

1/2

W[1,N ] (s, u) ds.

We then note that the RHS of the above inequality is bounded by the sum of the last two products on RHS (9.4.1) as desired. This completes our proof sketch of (9.4.1).

Shock formation for quasilinear wave systems featuring multiple speeds 88 The estimate (9.4.2) can similarly be proved by taking the norm k · kL2 (Σut ) of both sides of inequality (8.3.4). were handled in the proof of [28, Lemma 14.8] exZ t All terms ≤N ~ 0 cept for the last term P W (t , u, ϑ) dt0 on RHS (8.3.4), which, by the arguments t0 =0

given in the previous paragraph, can be bounded in the norm k · kL2 (Σut ) by ≤ C˚ + Z t 1 1/2 C W[1,N ] (s, u) ds as desired. 1/2 s=0 µ? (s, u) 9.5. L2 bounds for less degenerate top-order error integrals. In the next lemma, we bound some up-to-top-order error integrals that appear in our energy estimates. As in the proof of Lemma 9.10, the proof relies on modified quantities, which are needed to avoid the loss of a derivative. However, the estimates of the lemma are much less degenerate than those of Lemma 9.10 because of the availability of a helpful factor of µ in the integrands. Lemma 9.11 (Bounds for less degenerate top-order error integrals). Assume that 1 ≤ N ≤ 18 and let ρ be the scalar function from (2.8.7a). We have the following following integral estimates: Z # N N −1 ˘ ˘ ρ(XY Ψ)(XΨ)(d/ Ψ) · (µd/Y trg/ χ) d$ (9.5.1a) Mt,u Z t Z t 2 −1 0 0 0 . ln µ? (t , u) + 1 Q[1,N ] (t , u) dt + W[1,N ] (t0 , u) dt0 0 0 t =0 t =0 Z u Q[1,N ] (t, u0 ) du0 + ˚ 2 , + 0 Z u =0 ˘ (1 + 2µ)(LY N Ψ)(XΨ)(d /# Ψ) · (µd/Y N −1 trg/ χ) d$ , (9.5.1b) Mt,u Z ˘ (1 + 2µ)ρ(LY N Ψ)(XΨ)(d /# Ψ) · (µd/Y N −1 trg/ χ) d$ Mt,u Z t Z t 2 0 0 −1 0 . ln µ? (t , u) + 1 Q[1,N ] (t , u) dt + W[1,N ] (t0 , u) dt0 t0 =0 t0 =0 Z u + Q[1,N ] (t, u0 ) du0 + ˚ 2 . u0 =0

Proof. See Subsect. 5.2 for some comments on the analysis. To prove (9.5.1b), we use the fact that ρ = f(γ)γ (see (2.12.2c)), the L∞ estimates of Prop. 5.6, Cauchy-Schwarz, and (9.2.5) to deduce Z Z N 2 N LY Ψ d$ + µY trg/ χ 2 d$ LHS (9.5.1b) . (9.5.2) Mt,u u

Mt,u

Z t

N

N 2 0

µY trg/ χ 2 2 u dt0 . LY Ψ L2 (P t ) du + L (Σt0 ) u0 u0 =0 t0 =0 Z u Z t

N

µY trg/ χ 2 2 u dt0 . . Q[1,N ] (t, u0 ) du0 + L (Σ ) Z

u0 =0

t0 =0

t0

J. Speck

89

To complete the proof of (9.5.1b), we must handle the final integral on RHS (9.5.2). To

2 bound the integral by ≤ RHS (9.5.1b), we bound the integrand µY N trg/ χ L2 (Σu ) by using t0

inequality (9.4.2) (with t0 in place of t), simple estimates of the form ab . a2 + b2 , and we in addition use (7.2.5) and the fact that Q[1,N ] is increasing in its arguments to bound the first time integral on RHS (9.4.2) as follows: Z

t0

s=0

1/2 0 1 1/2 0 Q[1,N ] (s, u) ds . ln µ−1 ? (t , u) + 1 Q[1,N ] (t , u). µ? (s, u)

Similarly, with the help of (7.2.6), we bound the second time integral on RHS (9.4.2) as follows: Z t0 1 1/2 1/2 W[1,N ] (s, u) ds . W[1,N ] (t0 , u). 1/2 s=0 µ? (s, u) Z t

N

µY trg/ χ 2 2 u dt0 are . RHS (9.5.1b), These steps yield that all terms generated by L (Σ ) t0

t0 =0

except for the following integral generated by the last term on RHS (9.4.2): 2

Z

t

˚

t0 =0

0 ln µ−1 ? (t , u) + 1

2

dt0 .

Using (7.2.6), we deduce that the above term is . ˚ 2 as desired. We have thus proved (9.5.1b). The proof of (9.5.1a) starts with the following analog of (9.5.2), which can proved in the same way: Z LHS (9.5.1a) . .

t

t0 =0 Z t t0 =0

˘ N 2

XY Ψ

L2 (Σu ) t0

0

Z

0

0

Q[1,N ] (t , u) dt +

t

dt + t0 =0

Z

t

t0 =0

N

µY trg/ χ 2 2 u dt0 L (Σ ) t0

(9.5.3)

N

µY trg/ χ 2 2 u dt0 . L (Σ ) t0

The remaining details are similar to the ones that we gave above in our proof of (9.5.1b); we therefore omit them. 9.6. Estimates involving simple error terms. In this subsection, we derive L2 bounds for some simple error terms that encounter in the energy estimates. We start with a lemma in which we control the L2 norms of the easy derivatives of the eikonal function quantities, that is, for the derivatives that do not require the modified quantities described in Subsubsect. 1.7.6. Lemma 9.12 (L2 bounds for the eikonal function quantities that do not require modified quantities). The following estimates hold for 1 ≤ N ≤ 18 (see Subsect. 3.2

Shock formation for quasilinear wave systems featuring multiple speeds 90 regarding the vectorfield operator notation): 1/2

Q[1,N ] (t, u)

≤N −1

LP∗[1,N ] µ 2 u , LP ≤N Li(Small) ˚ , LP tr χ . + , g / u 1/2 L2 (Σt ) L (Σt ) L2 (Σu t) µ? (t, u) (9.6.1a) 1/2

Q[1,N ] (t, u)

LZ ≤N −1;1 trg/ χ 2 u . ˚

LZ ≤N ;1 Li(Small) , + , u 1/2 L (Σt ) L2 (Σt ) µ? (t, u) (9.6.1b)

≤N −1

[1,N ]

P

2 u .˚

P∗ µ 2 u , P ≤N Li(Small) , tr χ + g / u 2 L (Σ ) L (Σ ) L (Σ ) t

t

t

Z t Q1/2 (s, u) [1,N ] 1/2 s=0 µ? (s, u)

ds,

(9.6.1c)

≤N ;1 i

Z∗ L(Small) L2 (Σu ) , Z ≤N −1;1 trg/ χ L2 (Σu ) . ˚ + t

t

t Q1/2 (s, u) [1,N ] 1/2 s=0 µ? (s, u)

Z

ds.

(9.6.1d) Proof. Thanks to the estimates of Props. 5.6 and 7.3 and Lemma 9.3, the proof of [28, Lemma 14.3] goes through verbatim. In particular, these estimates do not involve the slow ~. wave variable W We now estimate the error integrals generated by the terms Harmless≤N and Harmless≤N (Slow) on the RHSs of the equations of Prop. 8.1. Lemma 9.13 (L2 bounds for error integrals involving Harmless≤N and Harmless≤N (Slow) terms). Assume that 1 ≤ N ≤ 18 and ς > 0. Recall that we defined terms of the form Harmless≤N and Harmless≤N (Slow) in Def. 8.1. We have the following estimates, where the implicit constants are independent of ς: (1 + µ)LP N Ψ Z Harmless≤N ˘ NΨ d$ XP (9.6.2) Harmless≤N (Slow) Mt,u [1,N ] ~ W P Z t Z u −1 0 0 −1 . (1 + ς ) Q[1,N ] (t , u) dt + (1 + ς ) Q[1,N ] (t, u0 ) du0 t0 =0 u0 =0 Z u + ςK[1,N ] (t, u) + (1 + ς −1 ) W[1,N ] (t, u0 ) du0 + ˚ 2 . u0 =0

Proof. See Subsect. 5.2 for some comments on the analysis. The bounds for error integrals ~ can be derived by using arguments identical to the ones given in the that do not involve W proof of [28, Lemma 14.7]; we therefore omit those details. ~ , which means that we must It remains for us to prove the desired bounds involving W estimate the spacetime integrals of various quadratic terms. In this proof, we derive the desired estimates only for some representative quadratic terms. The remaining terms can be similarly bounded and we omit those details. Specifically, we show how bound the integral [1,N ] ~ ≤N +1;1 of the product P W Z∗ Ψ . We note that our proof of the desired bound relies on

J. Speck

91

all of the main ideas needed to prove all of the estimates stated in the lemma. To proceed, we repeatedly use the commutation estimate (5.6.5) and the L∞ estimates of Prop. 5.6 ˘ in the operator Z∗≤N +1;1 so that it acts last, to commute the (at most one) factor of X ˘ [1,N ] Ψ| + |P ≤N +1 Ψ| + |Z∗≤N ;1 γ| + |P∗[1,N ] γ|. Thus, we which yields |Z∗≤N +1;1 Ψ| . |XP must bound the integral of the four corresponding products generated by the RHS of the previous inequality. To bound the integral of the first product, we use Young’s inequality and Lemma 9.4 to deduce Z [1,N ] ~ ˘ [1,N ] W XP Ψ d$ (9.6.3) P Mt,u Z u

.

Put 0

u0 =0

Z [1,N ] ~ 2 0 W d$ du + P

u0 =0

t

Z Σu t0

t0 =0

u

Z .

Z

Q[1,N ] (t, u0 ) du0 +

Z

˘ [1,N ] 2 Ψ d$ dt0 XP

t

t0 =0

Q[1,N ] (t0 , u) dt0 ,

which is . RHS (9.6.2) as desired. ~ ||P ≤N +1 Ψ|, we first consider terms of the form To handle the second product |P [1,N ] W ≤N [1,N ] ~ W ||Y P Ψ|. To bound the corresponding integrals, we use Young’s inequality and |P Lemma 9.4 and consider separately the regions in which µ < 1/4 and µ > 1/4 to deduce Z

[1,N ] ~ W Y P ≤N Ψ d$ P Mt,u Z u Z Z [1,N ] ~ 2 0 −1 W d$ du + ς . (1 + ς ) P Put 0

u0 =0

Z

t

Z

+ t0 =0

. (1 + ς

Σu t0 −1

Z

t

t0 =0

Z

µ |Y Ψ|2 d$ dt0

Σu t0

u 0

) u0 =0

2 1{µ≤1/4} Y P [1,N ] Ψ d$

Mt,u

2 µ d/P [1,N ] Ψ d$ dt0 +

Z

(9.6.4)

Z

0

Q[1,N ] (t, u ) du + ςK[1,N ] (t, u) +

t

t0 =0

Q[1,N ] (t0 , u) dt0 + ˚ 2 ,

which is . RHS (9.6.2) as desired. ~ ||P ≤N +1 Ψ|, it reTo finish the proof of the desired bounds for the second product |P [1,N ] W ~ ||LP ≤N Ψ| and of the form |P [1,N ] W ~ ||Ψ|. mains for us to consider terms of the form |P [1,N ] W We can bound the spacetime integrals of both of these terms by using arguments similar to Z the ones we used in proving (9.6.3), except that we also rely on the bounds |LP [1,N ] Ψ|2 d$ . Mt,u Z u Z Z u Z Z t 0 0 2 0 0 2 Q[1,N ] (t, u ) du , |LΨ| d$ . Q[1,N ] (t, u ) du , and |Ψ| d$ . Q1 (t0 , u) dt0 + u0 =0 2

Mt,u

u0 =0

Mt,u

t0 =0

˚ , which are simple consequences of Lemma 9.4. ~ ||Z ≤N ;1 γ| and the fourth product To bound the integral of the third product |P [1,N ] W ∗ [1,N ] [1,N ] ~ |P W ||P∗ γ|, we use arguments similar to the ones we used above as well as the estimates (9.6.1c) and (9.6.1d) which, when combined with the estimate (7.2.6) and the fact

Shock formation for quasilinear wave systems featuring multiple speeds 92 that Q[1,N ] is increasing in its arguments, yield the following spacetime integral bounds: Z Z [1,N ] 2 ≤N ;1 i 2 P∗ µ d$ Z∗ (9.6.5) L(Small) d$ + Mt,u

Mt,u

Z .

t0 =0

Z .

t

(Z

)2

1/2

t0

Q[1,N ] (s, u)

s=0

µ? (s, u)

1/2

ds

dt0 + ˚ 2

t

t0 =0

Q[1,N ] (t0 , u) dt0 + ˚ 2 ,

which is . RHS (9.6.2) as desired. This completes our proof of the representative estimates. 9.7. Estimates for the error terms not depending on inhomogeneous terms. In the next lemma, we derive bounds for the error integrals whose integrands were pointwise bounded in Lemma 8.3. Lemma 9.14 (Estimates for the error integrals not depending on inhomogeneous terms). Assume that 1 ≤ N ≤ 18. Let ς > 0 be a real number. We have the following estimate for the last term on RHS (4.2) (with P N Ψ in the role of f and without any absolute value taken on the left), where the implicit constants are independent of ς: Z

5 X

(T )

Z

t

1 p Q[1,N ] (t0 , u) dt0 (9.7.1) 0 0 T(Boot) − t t =0 Z t −1 Q[1,N ] (t0 , u) dt0 + (1 + ς ) 0 Zt u=0 + (1 + ς −1 ) Q[1,N ] (t, u0 ) du0 + ςK[1,N ] (t, u).

N

P(i) [P Ψ] d$ .

Mt,u i=1

u0 =0

Moreover, we have the following estimate for the term on the last line of RHS (4.3.4) (with ~ in the role of V~ ): PN W Z Z u N ~ W[1,N ] (t, u0 ) du0 . (9.7.2) {1 + γf(γ)} W[P W ] d$ ≤ C u0 =0

Mt,u

Proof. See Subsect. 5.2 for some comments on the analysis. To prove (9.7.1), we integrate (8.4.1) (with P N Ψ in the role of Ψ) over Mt,u and use Lemma 9.4. ~ in the role of V~ ) and To prove (9.7.2), we use the pointwise bound (8.4.2) (with P N W the coerciveness estimate (9.2.8) to deduce that Z u Z u N ~ 2 0 LHS (9.7.2) . kP W kP t 0 du . W[1,N ] (t, u0 ) du0 (9.7.3) u0 =0

as desired.

u

u0 =0

J. Speck

93

9.8. Proof of Prop. 9.6. Armed with the estimates of the previous subsections, we are now ready to prove Prop. 9.6. Proof of (9.3.1a): Assume that 1 ≤ N ≤ 18 and let P N be an N th -order Pu -tangent vectorfield operator. From (4.3.1) with P N Ψ in the role of f and definition (9.1.2a), we have E(F ast) [P N Ψ](t, u) + F(F ast) [P N Ψ](t, u) + K[P N Ψ](t, u)

(9.8.1)

= E(F ast) [P N Ψ](0, u) + E(F ast) [P N Ψ](t, 0) Z n o ˘ N Ψ µg (P N Ψ) d$ − (1 + 2µ)(LP N Ψ) + 2XP Mt,u

+

5 Z X i=1

(T )

P(i) [P N Ψ] d$.

Mt,u

~ in the role of V~ , we have Similarly, if 1 ≤ N ≤ 18, then by (4.3.4) with P N W ~ ](t, u) + F(Slow) [P N W ~ ](t, u) E(Slow) [P N W (9.8.2) ~ ](0, u) + F(Slow) [P N W ~ ](t, 0) = E(Slow) [P N W Z ~ ] d$ + {1 + γf(γ)} W[P N W M Z t,u + {1 + γf(γ)} 4(h−1 )α0 (h−1 )β0 (P N wα )Fβ + 2(h−1 )αβ (P N wα )Fβ d$ M Z t,u + {1 + γf(γ)} 2(h−1 )αa (h−1 )b0 (P N wα )Fab + 2(P N w)F d$, Mt,u

where F0 , Fa , F , and Fab respectively denote the terms Harmless≤N + Harmless≤N (Slow) on RHSs (8.2.2a)-(8.2.2d). We will show that RHS (9.8.1) ≤ RHS (9.3.1a) and that RHS (9.8.2) ≤ RHS (9.3.1a). Then, taking the appropriate maxes over these estimates and appealing to Defs. 9.1 and 9.2, we conclude (9.3.1a). We now show that RHS (9.8.1) ≤ RHS (9.3.1a). The vast majority of the terms on RHS (9.8.1) were suitably bounded by ≤ RHS (9.3.1a) in [28, Section 14.8]. In particular, the terms on the first line of RHS (9.8.1) were bounded by . ˚ 2 in Lemma 9.5 while the last integral on RHS (9.8.1) was bounded via the estimate (9.7.1), which are precisely the same estimates proved in [28]. We now explain the origin of the new terms and how to bound them. We stress that all of the new terms are non-borderline in the sense that they do not contribute to the “boxed-constant-involving” terms on RHS (9.3.1a), which drive the blowup-rate of the toporder energies. The new terms are all found within the factor µg (P N Ψ) in the first error R integral − Mt,u · · · on RHS (9.8.1). To bound this error integral, we first use (depending on the structure of P N ) one of equations (8.2.1a)-(8.2.1b) or (8.2.1c) to algebraically substitute for µg (P N Ψ). The error integrals generated by the Harmless≤N terms were suitably bounded in [28, Section 14.8] (see also Lemma 9.13). To bound the error integrals generated by the terms Harmless≤N (Slow) , we use Lemma 9.13. It remains for us to bound the error integrals generated by the three products on RHSs (8.2.1a)-(8.2.1b) that are not of the form

Shock formation for quasilinear wave systems featuring multiple speeds 94 Harmless≤N + Harmless≤N (Slow) . We proceed on a case by case basis, depending on the structure of the commutator vectorfield string P N . In the case P N = Y N , we must bound the difficult error integral Z N ˘ N Ψ)(XΨ)Y ˘ −2 (XY trg/ χ d$, (9.8.3) Mt,u N ˘ which is generated by the term (XΨ)Y trg/ χ on RHS (8.2.1b), by ≤ RHS (9.3.1a). To this end, we first use Cauchy-Schwarz and (9.2.5) to bound the magnitude of (9.8.3) by Z t

˘

1/2 N ≤2 Q[1,N ] (t0 , u) (XΨ)Y trg/ χ dt0 . (9.8.4) u L2 (Σt0 )

t0 =0

We now substitute the estimate (9.4.1) (with t replaced by t0 ) for the second integrand factor on RHS (9.8.4). In [28, Section 14.8], all of the corresponding terms that arise from this substitution were bounded by ≤ RHS (9.3.1a) except for the term generated by the one product on RHS (9.4.1) involving W[1,N ] (that is, the next-to-last product on RHS (9.4.1)). Upon substituting this remaining product into (9.8.4), we generate the following error term, which is new compared to the terms appearing in [28, Section 14.8]: Z t0 Z t 1 1 1/2 1/2 0 Q[1,N ] (t , u) W[1,N ] (s, u) ds dt0 . (9.8.5) C 1/2 0 µ (t , u) 0 s=0 µ? (s, u) t =0 ? Note that the term (9.8.5) is ≤ the last product on RHS (9.3.1a) as desired. The error integral Z N ˘ − (LY N Ψ)(XΨ)Y trg/ χ d$, Mt,u N ˘ which is also generated by the term (XΨ)Y trg/ χ on RHS (8.2.1b), was bounded by ≤ ~ and therefore RHS (9.3.1a) in [28, Section 14.8] using arguments that are independent of W do not involve the slow wave controlling quantities WM . We do not repeat the proof here since it is exactly the same and since it relies on a lengthy integration by parts argument in which one trades the factor of L from LY N Ψ for a factor of Y from Y N trg/ χ. We note that the integration by parts generates some of the “boxed-constant-involving” terms on 1 1/2 RHS (9.3.1a), including the “boundary term” 2 1/2 Q[1,N ] (t, u) kLµkL∞ ((−)Σut;t ) · · · on µ? (t, u) the fourth line of RHS (9.3.1a). To bound the remaining three error integrals generated by RHS (8.2.1b), namely Z N ˘ − 2µ(LY N Ψ)(XΨ)Y trg/ χ d$, Mt,u

Z −2

˘ N Ψ)(d/# Ψ) · (µd/Y N −1 trg/ χ) d$, (XY

Mt,u

and

Z

(1 + 2µ)(LY N Ψ)(d/# Ψ) · (µd/Y N −1 trg/ χ) d$ Mt,u by ≤ RHS (9.3.1a), we use Lemma 9.11 as well as the simple bound Y N trg/ χ . d/Y N −1 trg/ χ (see point (5) of Subsect. 5.2). −

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To complete the proof that RHS (9.8.1) ≤ RHS (9.3.1a), it remains only for us to bound the difficult error integrals that arise in the case P N = Y N −1 L, which are generated by the terms on RHS (8.2.1a) that are not of the form Harmless≤N + Harmless≤N (Slow) . Specifically, we encounter the error integrals Z ˘ N −1 LΨ)(d/# Ψ) · (µd/Y N −1 trg/ χ) d$, −2 (XY Mt,u

and

Z −

(1 + 2µ)(LY N −1 LΨ)(d/# Ψ) · (µd/Y N −1 trg/ χ) d$,

Mt,u

which we bounded in Lemma 9.11. We have thus shown that RHS (9.8.1) ≤ RHS (9.3.1a). To complete the proof of (9.3.1a), it remains only for us to show that RHS (9.8.2) ≤ ~ ](0, u)+F(Slow) [P N W ~ ](t, 0) . RHS (9.3.1a). First, from Lemma 9.5, we obtain E(Slow) [P N W 2 ˚ as desired. Next, we note that we handled the first integral on RHS (9.8.2) in Lemma 9.14. To bound the integrals on the last two lines of RHS (9.8.2), we recall that, as we mentioned above, the terms F0 , Fa , F , and Fab in the integrands are of the form Harmless≤N + Harmless≤N Moreover, the L∞ estimates of Prop. 5.6 imply that |1 + γf(γ)| . 1 (Slow) . ~ ), that (h−1 )αβ . 1. Thus, the desired bounds follow from and, since (h−1 )αβ = f(γ, W Lemma 9.13. This completes the proof of (9.3.1a). Proof of (9.3.1b) The proof of (9.3.1b) is similar to that of (9.3.1a) but involves one key change. To proceed, we let P N −1 be an (N − 1)st -order Pu -tangent vectorfield operator, where 2 ≤ N ≤ 18. We then argue as above, starting with the identity (9.8.1) with P N −1 Ψ ~ . We bound almost ~ in place of P N W in place of P N Ψ and the identity (9.8.2) with P N −1 W all error integrals in exactly the same way as before, the key change being that we bound the two most difficult error integrals in a different way. Specifically, the two difficult integrals are Z N −1 ˘ N −1 Ψ)(XΨ)Y ˘ −2 (XY trg/ χ d$, Mt,u

which is an analog of (9.8.3), and Z N −1 ˘ − (LY N −1 Ψ)(XΨ)Y trg/ χ d$, Mt,u

whose top-order analog we did not discuss in detail above since it was treated in [28, Section ~ . Both of the 14.8] using arguments that are independent of the slow wave variable W above error integrals were shown in [28, Section 14.8] to be bounded by ≤ RHS (9.3.1b) using a simple argument that does not involve the slow wave controlling quantities WM ; the key point is to use the derivative-losing estimate (9.6.1c) to control kY N −1 trg/ χkL2 (Σut ) . We remark that the arguments in [28, Section 14.8] for bounding these two error integrals lead 1/2 to the presence of the Q[1,N ] -involving double time integral on RHS (9.3.1b). That is, these estimates lose one derivative (which is permissible below top-order) and thus are coupled to the next-highest-order energy estimates; the gain is that the resulting integrals are less singular with respect to µ−1 ? . We have thus proved Prop. 9.6.

Shock formation for quasilinear wave systems featuring multiple speeds 96 10. The main stable shock formation theorem We now prove our main result. ~ to the system (1.6.2a) Theorem 10.1 (Stable shock formation). Consider a solution Ψ, W + (1.6.11a)-(1.6.11d) with nonlinearities verifying the assumptions stated in Subsubsects. 1.6.1(1.6.3). Let ˚ , ˚ δ, and ˚ δ∗ be the data-size parameters introduced in Sect. 3. For each U0 ∈ [0, 1], let T(Lif espan);U0 be the classical lifespan of the solution in the region that is completely determined by the non-trivial data lying in ΣU0 0 and the small data given along ˚−1 P02δ∗ (see Figure 2 on pg. 23). If ˚ is sufficiently small relative to ˚ δ−1 and ˚ δ∗ (in the sense explained in Subsect. 3.8), then the following conclusions hold, where all constants can be chosen to be independent of U0 . Dichotomy of possibilities. One of the following mutually disjoint possibilities must occur, where µ? (t, u) = min{1, minΣut µ}. I) T(Lif espan);U0 > 2˚ δ−1 ∗ . In particular, the solution exists classically on the spacetime region clM2˚δ−1 , where cl denotes closure. Furthermore, inf{µ? (s, U0 ) | s ∈ ∗ ,U0 −1 [0, 2˚ δ∗ ]} > 0. II) T(Lif espan);U0 ≤ 2˚ δ−1 ∗ , and n o T(Lif espan);U0 = sup t ∈ [0, 2˚ δ−1 ) | inf{µ (s, U ) | s ∈ [0, t)} > 0 . (10.0.1) ? 0 ∗ In addition, case II) occurs when U0 = 1. In this case, we have T(Lif espan);1 = {1 + O(˚ )} ˚ δ−1 ∗ .

(10.0.2)

What happens in Case I). In case I), all of the bootstrap assumptions from Subsects. 3.53.6, the estimates of Props. 5.6 and 6.2, and the energy estimates of Prop. 9.7 hold on with the factors of ε on the RHSs replaced by C˚ . Moreover, for 0 ≤ M ≤ 5, clM2˚δ−1 ∗ ,U0 −1 ˚ the following estimates hold for (t, u) ∈ [0, 2δ∗ ] × [0, U0 ] (see Subsect. 3.2 regarding the vectorfield operator notation):

[1,12]

≤11

P∗ µ 2 u , P ≤12 Li(Small)

P trg/ χ 2 u ≤ C˚ , , (10.0.3a) u L (Σt ) L (Σt ) L2 (Σt )

13+M

13+M i

12+M

P∗ µ?−(M +.4) (t, u), µ L2 (Σu ) , P L(Small) L2 (Σu ) , P trg/ χ L2 (Σu ) ≤ C˚ t t t (10.0.3b)

LP 18 µ 2 u , LZ 18;1 Li(Small) (t, u), (10.0.3c) , LZ 17;1 trg/ χ L2 (Σu ) ≤ C˚ µ−6.4 ? L (Σt ) L2 (Σu t t)

18

µY trg/ χ 2 u ≤ C˚ µ−5.9 (t, u). (10.0.3d) ? L (Σ ) t

What happens in Case II). In case II), all of the bootstrap assumptions from Subsects. 3.5-3.6, the estimates of Props. 5.6 and 6.2, and the energy estimates of Prop. 9.7 hold on MT(Lif espan);U0 ,U0 with the factors of ε on the RHS replaced by C˚ . Moreover, for 0 ≤ M ≤ 5, the estimates (10.0.3a)-(10.0.3d) hold for (t, u) ∈ [0, T(Lif espan);U0 ) × [0, U0 ]. In ˘X ˘ XΨ, ˘ Z ≤9;1 W ~,X ˘X ˘W ~ , Z ≤9;1 Li , X ˘ XL ˘ i, addition, the scalar functions Z ≤9;1 Ψ, Z ≤4;2 Ψ, X U ≤9 ≤2;1 0 ˘ Xµ ˘ extend to Σ P µ, Z µ, and X T(Lif espan);U as functions of the geometric coordinates 0

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97

(t, u, ϑ) that are uniformly bounded in L∞ (Σut ) for (t, u) ∈ [0, T(Lif espan);U0 ] × [0, U0 ]. Furthermore, the Cartesian component functions gαβ (Ψ) verify the estimate gαβ = mαβ + O(˚ ) (where mαβ = diag(−1, 1, 1) is the standard Minkowski metric) and have the same extension properties as Ψ and its derivatives with respect to the vectorfields mentioned above. U0 ;(Blowup) 0 Moreover, let ΣT(Lif be the (non-empty) subset of ΣUT(Lif defined by espan);U0 espan);U0 U0 ;(Blowup) ΣT(Lif := (T , u, ϑ) | µ(T , u, ϑ) = 0 . (10.0.4) (Lif espan);U (Lif espan);U 0 0 espan);U 0

U ;(Blowup)

0 , there exists a past neighborhood Then for each point (T(Lif espan);U0 , u, ϑ) ∈ ΣT(Lif espan);U0 containing it such that the following lower bound holds in the neighborhood: ˚ δ∗ 1 |XΨ(t, u, ϑ)| ≥ . (10.0.5) 4|GL(F lat) L(F lat) (Ψ = 0)| µ(t, u, ϑ)

In (10.0.5),

˚ δ∗ is a positive data-dependent constant (see (1.6.9)), 4 GL(F lat) L(F lat) (Ψ = 0)

and the `t,u -transversal vectorfield X is near-Euclidean-unit length: δab X a X b = 1 + O(˚ ). In U0 ;(Blowup) particular, XΨ blows up like 1/µ at all points in ΣT(Lif espan);U . Conversely, at all points in 0

U ;(Blowup)

0 0 (T(Lif espan);U0 , u, ϑ) ∈ ΣUT(Lif \ΣT(Lif , we have espan);U0 espan);U0 XΨ(T(Lif espan);U0 , u, ϑ) < ∞.

(10.0.6)

Discussion of proof. The proof of [28, Theorem 15.1] applies nearly verbatim, except for a ~ . The main ingredients in the proof are the estimates of few of the statements regarding W Props. 5.6 and 6.2, the energy estimates of Prop. 9.7, and Cor. 9.9. Strictly speaking, in the proof of [28, Theorem 15.1], a few additional estimates, beyond the main ingredients we just mentioned, were needed to complete the proof. For example, one needs estimates for all of the components of the change of variables map Υ, including the scalar-valued functions Ξi on RHS (2.7.1). However, in [28], the needed estimates followed as straightforward consequences of the main ingredients mentioned above, and we do not even bother to state them here since their proofs are exactly the same as in [28]; in particular the proofs of the omitted estimates ~. do not involve W ~ that are not part of the statement For completeness, we now prove some facts concerning W ~ and X ˘X ˘W ~ extend to or proof of [28, Theorem 15.1]. Specifically, the facts that Z ≤9;1 W U0 ΣT(Lif espan);U as L∞ as functions of the geometric coordinates (t, u, ϑ) that are uniformly 0 u bounded in L∞ (Σ

∈ [0, T (Lif espan);U 0 ] × [0, U0 ] are simple consequences of the

t ) for (t, u)

˘ ˘~

≤9;1 ~ , LX XW . 1, which are special cases of the uniform bounds LZ W L∞ (Σu t)

L∞ (Σu t)

∂ estimates (5.7.4) and (6.2.4) (recall that L = ). ∂t

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