Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation

Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation

Jared Speck
Abstract.

In spatial dimensions, we study the Cauchy problem for a quasilinear transport equation coupled to a quasilinear symmetric hyperbolic subsystem of a rather general type. For an open set (relative to a suitable Sobolev topology) of regular initial data that are close to the data of a simple plane wave, we give a sharp, constructive proof of shock formation in which the transport variable remains bounded but its first-order Cartesian coordinate partial derivatives blow up in finite time. Moreover, we prove that the singularity does not propagate into the symmetric hyperbolic variables: they and their first-order Cartesian coordinate partial derivatives remain bounded, even though they interact with the transport variable all the way up to its singularity. The formation of the singularity is tied to the finite-time degeneration, relative to the Cartesian coordinates, of a system of geometric coordinates adapted to the characteristics of the transport operator. Two crucial features of the proof are that relative to the geometric coordinates, all solution variables remain smooth, and that the finite-time degeneration coincides with the intersection of the characteristics. Compared to prior shock formation results in more than one spatial dimension, in which the blowup occurred in solutions to wave equations, the main new features of the present work are: i) we develop a theory of nonlinear geometric optics for transport operators, which is compatible with the coupling and which allows us to implement a quasilinear geometric vectorfield method, even though the regularity properties of the corresponding eikonal function are less favorable compared to the wave equation case and ii) we allow for a full quasilinear coupling, i.e., the principal coefficients in all equations are allowed to depend on all solution variables.


Keywords: characteristics, eikonal equation, eikonal function, simple wave, vectorfield method, wave breaking


Mathematics Subject Classification (2010) Primary: 35L67 - Secondary: 35L45

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. jspeck@math.mit.edu

]0 Transport]J. Speck 0] ]

July 6, 2019

1. Introduction

The study of quasilinear hyperbolic PDE systems is one of the most classical pursuits in mathematics and, at the same time, among the most active. Such systems are of intense theoretical interest, in no small part due to the fact that their study lies at the core of the revered field of nonlinear hyperbolic conservation laws (more generally “balance laws”); we refer readers to Dafermos’ work [cD2010] for a detailed discussion of the history of nonlinear hyperbolic balance laws as well as a comprehensive introduction to the main results of the field and the main techniques behind their proofs, with an emphasis on the case of one spatial dimension. The subject of quasilinear hyperbolic systems is of physical interest as well, since they are used to model a vast range of physical phenomena. A fundamental issue surrounding the study of the initial value problem for such PDEs is that solutions can develop singularities in finite time, starting from regular initial data. In one spatial dimension, the theory is in a rather advanced state, and in many cases, the known well-posedness results are able to accommodate the formation of shock singularities as well as their subsequent interactions; see the aforementioned work of Dafermos. The advanced status of the one-space-dimensional theory is highly indebted to the availability of estimates in the space of functions of bounded variation (BV). In contrast, Rauch [jR1986] showed that for quasilinear hyperbolic systems in more than one spatial dimension, well-posedness in BV class generally does not hold. For this reason, energy estimates in -based Sobolev spaces play an essential role in multiple spatial dimensions, and even the question of whether or not there is stable singularity formation (starting from regular initial data) can be exceptionally challenging. In particular, in order to derive a constructive shock formation result in more than one spatial dimension, one cannot avoid the exacting task of deriving energy estimates that hold up to the singularity.

In view of the above remarks, it is not surprising that the earliest blowup results for quasilinear hyperbolic PDEs in more than one spatial dimension without symmetry assumptions were not constructive, but were instead based on proofs by contradiction, with influential contributions coming from, for example, John [fJ1981] for a class of wave equations and Sideris for a class of hyperbolic systems [tS1984] and later for the compressible Euler equations [tS1985]. The main idea of the proofs was to show that for smooth solutions with suitable initial data, certain spatially averaged quantities verify ordinary differential inequalities that force them to blow up, contradicting the assumption of smoothness.

Although the blowup results mentioned in the previous paragraph are compelling, their chief drawback is that they provide no information about the nature of the singularity, other than an upper bound on the solution’s classical lifespan. In particular, such results are not useful if one aims to extract sharp information about the blowup-mechanism and blowup-time, or if one aims to uniquely continue the solution past the singularity in a weak sense. In contrast, many state-of-the-art blowup-results for hyperbolic PDEs yield a detailed description of the singularity formation, even in the challenging setting of more than one spatial dimension. This is especially true for results on the formation of shocks starting from smooth initial conditions, a topic that has enjoyed remarkable progress in the last decade, as we describe in Subsect. 1.7. Our main results are in this vein. We recall that a shock singularity111The formation of a shock is sometimes referred to as “wave breaking.” is such that some derivative of the solution blows up in finite time while the solution itself remains bounded. Shock singularities are of interest in part due to their rather mild nature, which leaves open the hope that one might be able to extend the solution uniquely past the shock, in a weak sense, under suitable selection criteria. In the case of the compressible Euler equations in three spatial dimension, this hope has been realized in the form of Christodoulou’s recent breakthrough resolution [dC2017] of the restricted shock development problem without symmetry problems; see Subsubsect. 1.7.2 for further discussion.

We now provide a very rough statement of our results; see Theorem 1.5 on pg. 1.5 for a more detailed summary and Theorem 10.1 on pg. 10.1 for the complete statements.

Theorem 1.1 (Stable shock formation (very rough version)).

In an arbitrary number of spatial dimensions, there are many quasilinear hyperbolic PDE systems comprising a transport equation coupled to a symmetric hyperbolic subsystem such that the following occurs: there exists an open set of initial data without symmetry assumptions such that the transport variable remains bounded but its first derivatives blow up in finite time. More precisely, the derivatives of the transport variable in directions tangent to the transport characteristics remain bounded, while any derivative in a transversal direction blows up. Moreover, the singularity does not propagate into the symmetric hyperbolic variables; they remain bounded, as do their first derivatives in all directions.

Remark 1.2 (Rescaling the transversal derivative so as to “cancel” the blowup).

We note already that a key part of the proof is showing the derivative of the transport variable in the transversal direction also remains bounded. This does not contradict Theorem 1.1 for the following reason: the vectorfield is constructed so that its Cartesian components go to as the shock forms, in a manner that exactly compensates for the blowup of an “order-unity-length” transversal derivative of the transport variable. Roughly, the situation can be described as follows, where is the transport variable and the remaining quantities will be rigorously defined later in the article: blows up,222Here and throughout, if is a vectorfield and is a scalar function, then is the derivative of in the direction . remains bounded, , and the weight vanishes for the first time at the shock; one could say that blows up like as , where is the size of at the shock; see Subsubsect. 1.6.4 for a more in-depth discussion of this point.

Remark 1.3 (The heart of the proof and the kind of initial data under study).

The heart of the proof of Theorem 1.1 is to control the singular terms and to show that the shock actually happens, i.e., that chaotic interactions do not prevent the shock from forming or cause a more severe kind of singularity. In an effort to focus only on the singularity formation, we have chosen to study the simplest non-trivial set of initial data to which our methods apply: perturbations of the data corresponding to simple plane symmetric waves (see Subsect. 1.4 for further discussion), where we assume plentiful initial Sobolev regularity. The corresponding solutions do not experience dispersion, so there are no time or radial weights in our estimates. We will describe the initial data in more detail in Subsubsect. 1.6.3.

Remark 1.4 (Extensions to other kinds of hyperbolic subsystems).

From our proof, one can infer that the assumption of symmetric hyperbolicity for the subsystem from Theorem 1.1 is in itself not important; we therefore anticipate that similar shock formation results should hold for systems comprising quasilinear transport equations coupled to many other types of hyperbolic subsystems, such as wave equations or regularly hyperbolic (in the sense of [dC2000]) subsystems.

1.1. Paper outline

  • In the remainder of Sect. 1, we give a more detailed description of our main results, summarize the main ideas behind the proofs, place our work in context by discussing prior works on shock formation, and summarize some of our notation.

  • In Sect. 2, we precisely define the class of systems to which our main results apply.

  • In Sect. 3, we construct the majority of the geometric objects that play a role in our analysis. We also derive evolution equations for some of the geometric quantities.

  • In Sect. 4, we derive energy identities.

  • In Sect. 5, we state the number of derivatives that we use to close our estimates, state our size assumptions on the data, and state bootstrap assumptions that are useful for deriving estimates.

  • In Sect. 6, we derive pointwise estimates for solutions to the evolution equations and their derivatives, up to top order.

  • In Sect. 7, we derive some properties of the change of variables map from geometric to Cartesian coordinates.

  • In Sect. 8, which is the main section of the paper, we derive a priori estimates for all of the quantities under study.

  • In Sect. 9, we provide some continuation criteria that, in the last section, we use to show that the solution survives up to the shock.

  • In Sect. 10, we state and prove the main theorem.

1.2. The role of nonlinear geometric optics in proving Theorem 1.1

In prior stable shock formation results in more than one spatial dimension (which we describe in Subsubsect. 1.7.2), the blowup occurred in a solution to a wave equation. In the present work, the blowup occurs in the derivatives of the solution to the transport equation. The difference is significant in that to obtain the sharp picture of shock formation, one must rely on a geometric version of the vectorfield method that is precisely tailored to the family of characteristics whose intersection is tied to the blowup. The key point is that the basic regularity properties of the characteristics and the corresponding geometric vectorfields are different in the wave equation and transport equation cases; we will discuss this fundamental point in more detail below. Although the blowup mechanism for solutions to the transport equations under study is broadly similar to the Riccati-type mechanism that drives singularity formation in the simple one-space-dimensional example of Burgers’ equation333The Riccati term appears after one spatial differentiation of the equation. (see Subsect. 1.4 for related discussion), the proof of our main theorem is much more complicated, owing in part to the aforementioned difficulty of having to derive energy estimates in multiple spatial dimensions.

The overall strategy of our proof is to construct a system of geometric coordinates adapted to the transport characteristics, relative to which the solution remains smooth, in part because the geometric coordinates “hide”444In one spatial dimension, this is sometimes referred to as “straightening out the characteristics” via a change of coordinates. the Riccati-type term mentioned above. In more than one spatial dimension, the philosophy of constructing geometric coordinates to regularize the problem of shock formation seems to have originated Alinhac’s work [sA1999a, sA1999b, sA2001b] on quasilinear wave equations; see Subsubsect. 1.7.2 for further discussion. As will become abundantly clear, our construction of the geometric coordinates and other related quantities is tied to the following fundamental ingredient in our approach: our development of a theory of nonlinear geometric optics for quasilinear transport equations, tied to an eikonal function, that is compatible with full quasilinear coupling to the symmetric hyperbolic subsystem. We use nonlinear geometric optics to construct vectorfield differential operators adapted to the characteristics as well as to detect the singularity formation. By “compatible,” we mean, especially, from the perspective of regularity considerations. Indeed, in any situation in which one uses nonlinear geometric optics to study a quasilinear hyperbolic PDE system, one must ensure that the regularity of the corresponding eikonal function is consistent with that of the solution. By “full quasilinear coupling,” we mean that in the systems that we study, the principal coefficients in all equations are allowed to depend on all solution variables.

Upon introducing nonlinear geometric optics into the problem, we encounter the following key difficulty:

Some of the geometric vectorfields that we construct have Cartesian components that are one degree less differentiable than the transport variable, as we explain in Subsubsect. 1.6.5.

On the one hand, due to the full quasilinear coupling, it seems that we must use the geometric vectorfields when commuting the symmetric hyperbolic subsystem to obtain higher-order estimates; this allows us to avoid generating uncontrollable commutator error terms involving “bad derivatives” (i.e., in directions transversal to the transport characteristics) of the shock-forming transport variable. On the other hand, the loss of regularity of the Cartesian components of the geometric vectorfields leads, at the top-order derivative level, to commutator error terms in the symmetric hyperbolic subsystem that are uncontrollable in that they have insufficient regularity. To overcome this difficulty, we employ the following strategy:

We never commute the symmetric hyperbolic subsystem a top-order number of times with a pure string of geometric vectorfields; instead, we first commute it with a single Cartesian coordinate partial derivative, and then follow up the Cartesian derivative with commutations by the geometric vectorfields.

The above strategy allows us to avoid the loss of a derivative, but it generates commutator error terms depending on a single Cartesian coordinate partial derivative, which are dangerous because they are transversal to the transport characteristics. Indeed, the first Cartesian coordinate partial derivatives of the transport variable blow up at the shock. Fortunately, by using a weight555The weight is the quantity from Remark 1.2, and we describe it in detail below. adapted to the characteristics, we are able to control such error terms featuring a single Cartesian differentiation, all the way up to the singularity.

We close this subsection by providing some remarks on using nonlinear geometric optics to study the maximal development666The maximal development of the data is, roughly, the largest possible classical solution that is uniquely determined by the data. Readers can consult [jSb2016, wW2013] for further discussion. of initial data for quasilinear hyperbolic PDEs without symmetry assumptions. The approach was pioneered by Christodoulou–Klainerman in their celebrated proof [dCsK1993] of the stability of Minkowski spacetime as a solution to the Einstein-vacuum equations.777Roughly, [dCsK1993] is a small-data global existence result for Einstein’s equations. Since perturbative global existence results for hyperbolic PDEs typically feature estimates with “room to spare,” in many cases, it is possible to close the proofs by relying on a version of approximate nonlinear geometric optics, which features approximate eikonal functions whose level sets approximate the characteristics. The advantage of using approximate eikonal functions is that is that their regularity theory is typically very simple. For example, such an approach was taken by Lindblad–Rodnianski in their proof of the stability of the Minkowski spacetime [hLiR2010] relative to wave coordinates. Their proof was less precise than Christodoulou–Klainerman’s but significantly shorter since, unlike Christodoulou–Klainerman, Lindblad–Rodnianski relied on approximate eikonal functions whose level sets were standard Minkowski light cones.

The use of eikonal functions for proving shock formation for quasilinear wave equations in more than one spatial dimension without symmetry assumptions was pioneered by Alinhac in his aforementioned works [sA1999a, sA1999b, sA2001b], and his approach was later remarkably sharpened/extended by Christodoulou [dC2007]. In contrast to global existence problems, in proofs of shock formation without symmetry assumptions, the use of an eikonal function adapted to the true characteristics (as opposed to approximate ones) seems essential, since the results yield that the singularity formation exactly coincides with the intersection of the characteristics. One can also draw an analogy between works on shock formation and works on low regularity well-posedness for quasilinear wave equations, such as [sKiR2003, sKiR2005d, hSdT2005, sKiRjS2015], where the known proofs fundamentally rely on eikonal functions whose levels sets are true characteristics.

1.3. A more precise statement of the main results

For the systems under study, we assume that the number of spatial dimensions is , where is arbitrary. For convenience, we study the dynamics of solutions in spacetimes of the form , where

(1.3.1)

is the spatial manifold and is the standard dimensional torus (i.e., with the endpoints identified and equipped with the usual smooth orientation). The factor in (1.3.1) will correspond to perturbations away from plane symmetry. Our assumption on the topology of is for technical convenience only; since our results are localized in spacetime, one could derive similar stable blowup results for arbitrary spatial topology.888However, assumptions on the data that lead to shock formation generally must be adapted to the spatial topology. Throughout, are a fixed set of Cartesian spacetime coordinates on , where is the time coordinate, are the spatial coordinates on , is the “non-compact space coordinate,” and are standard locally defined coordinates on such that is a positively oriented frame. We denote the Cartesian coordinate partial derivative vectorfields by , and we sometimes use the alternate notation . Note that the vectorfields can be globally defined so as to form a smooth frame, even though the are only locally defined. For mathematical convenience, our main results are adapted to nearly plane symmetric solutions, where by our conventions, exact plane symmetric solutions depend only on and . We now roughly summarize our main results; see Theorem 10.1 for precise statements.

Theorem 1.5 (Stable shock formation (rough version)).

Assumptions: Consider the following coupled system999Throughout we use Einstein’s summation convention. Greek lowercase “spacetime” indices vary over , while Latin lowercase “spatial” indices vary over . with initial data posed on the constant-time hypersurface :

(1.3.2)
(1.3.3)

where is a scalar function, is an array ( is arbitrary), and the are symmetric matrices. Assume that verifies a genuinely nonlinear-type condition tied to its dependence on (specifically, condition (2.2.1)) and that for small and , the constant-time hypersurfaces and the are spacelike101010This means that is positive definite, where the one-form is co-normal to the surface and satisfies . for the subsystem (1.3.3). Here and throughout, the are -characteristics, which are the family of (solution-dependent) hypersurfaces equal to the level sets of the eikonal function , that is, the solution to the eikonal equation (see Footnote 2 regarding the notation) with the initial condition .

To close the proof, we make the following assumptions on the data, which we propagate all the way up to the singularity:

Along , , all of its derivatives, and the -tangential derivatives of are small relative111111We also assume an absolute smallness condition on . to quantities constructed out of a first -transversal derivative of (see Subsect. 5.4 for the precise smallness assumptions, which involve geometric derivatives). Moreover, along , all derivatives of up to top order are relatively small.

Conclusions: There exists an open set (relative to a suitable Sobolev topology) of data that are close to the data of a simple plane wave (where a simple plane wave is such that and ), given along the unity-thickness subset of and a finite portion of , such that the solution behaves as follows:

blows up in finite time while , , and remain uniformly bounded.

The blowup is tied to the intersection of the , which in turn is precisely characterized by the vanishing of the inverse foliation density of the , which is initially near unity; see Fig. 1 for a picture in which a shock is about to form (in the region up top, where is small). Moreover, one can complete to form a geometric coordinate system on spacetime with the following key property, central to the proof:

No singularity occurs in , , , or their derivatives with respect to the geometric coordinates121212In practice, we will derive estimates for the derivatives of the solution with respect to the vectorfields depicted in Fig. 1. up to top order.

Put differently, the problem of shock formation can be transformed into an equivalent problem in which one proves non-degenerate estimates relative to the geometric coordinates and, at the same time, proves that the geometric coordinates degenerate in a precise fashion with respect to the Cartesian coordinates as .

\setkeys

Ginscale=.25\OVP@calc

Figure 1. The dynamics until close to the time of the shock when
Remark 1.6 (Non trivial interactions all the way up to the singularity).

We emphasize that in Theorem 1.5, can be non-zero at the singularity in . This means, in particular, that the problem cannot be reduced to the study of blowup for the simple case of a decoupled scalar transport equation.

Remark 1.7 (Extensions to allow for semilinear terms).

We expect that the results of Theorem 1.5 could be extended to allow for the presence of arbitrary smooth semilinear terms on RHSs (1.3.2)-(1.3.3) that are functions of . The extension would be straightforward to derive for semilinear terms that vanish when (for example, ). The reason is that our main results imply that such semilinear terms remain small, in suitable norms, up to the shock. In fact, such semilinear terms completely vanish for the exact simple waves whose perturbations we treat in Theorem 1.5; see Subsect. 1.4 for further discussion of simple waves. Consequently, a set of initial data similar to the one from Theorem 1.5 would also lead to the formation of a shock in the presence of such semilinear terms. In contrast, for semilinear terms that do not vanish when (for example, ), the analysis would be more difficult and the assumptions on the data might have to be changed to produce shock-forming solutions. In particular, such semilinear terms can, at least for data with large, radically alter the behavior of some solutions. This can be seen in the simple model problem of the inhomogeneous Burgers-type equation . This equation admits the family of ODE-type blowup solutions , whose singularity is much more severe than the shocks that that typically form when the semilinear term is absent.

Remark 1.8 (Description of a portion of the maximal development).

We expect that the approach that we take in proving our main theorem is precise enough that it can be extended to yield sharp information about the behavior of the solution up the boundary of the maximal development, as Christodoulou did in his related work [dC2007]*Chapter 15 (which we describe in Subsubsect. 1.7.2). For brevity, we do not pursue this issue in the present article. However, in the detailed version of our main results (i.e., Theorem 10.1), we set the stage for the possible future study of the maximal development by proving a “one-parameter family of results,” indexed by ; one would need to vary to study the maximal development. Here and throughout, corresponds to an initial data region of thickness ; see Fig. 2 on pg. 2 and Subsubsect. 1.6.2 for further discussion. For , which is implicitly assumed in Theorem 1.5, a shock forms in the maximal development of the data given along131313Actually, as we explain in Subsubsect. 1.6.2, we only need to specify the data along the subset of . . However, for small , a shock does not necessarily form in the maximal development of the data given along within the amount of time that we attempt to control the solution.

1.4. Further discussion on simple plane symmetric waves

Theorem 1.5 shows, roughly, that the well-known stable blowup of in solutions to the one-space-dimensional Burgers’ equation

(1.4.1)

is stable under a full quasilinear coupling of (1.4.1) to other hyperbolic subsystems, under perturbations of the coefficients in the transport equation, and under increasing the number of spatial dimensions. We now further explain what we mean by this. A special case of Theorem 1.5 occurs when and depends only on and (plane symmetry). In this simplified context, the blowup of can be proved using a simple argument based on the method of characteristics, similar to the argument that is typically use to prove blowup in the case of Burgers’ equation. Solutions with are sometimes referred to as simple waves since they can be described by a single non-zero scalar component. From this perspective, we see that Theorem 1.5 yields the stability of simple plane wave blowup for the transport variable in solutions to the system (1.3.2)-(1.3.3).

1.5. The main new ideas behind the proof

The proof of Theorem 1.5 is based in part on ideas used in earlier works on shock formation in more than one spatial dimension. We review these works in Subsect. 1.7. Here we summarize the two most novel aspects behind the proof of Theorem 1.5.

  • (Nonlinear geometric optics for transport equations) As in all prior shock formation results in more than one spatial dimension, our proof relies on nonlinear geometric optics, that is, the eikonal function . The use of an eikonal function is essentially the method of characteristics implemented in more than one spatial dimension. All of the prior works were such that the blowup occurred in a solution to a quasilinear wave equation and thus the theory of nonlinear geometric optics was adapted to those wave characteristics. In this article, we advance the theory of nonlinear geometric optics for transport equations. Although the theory is simpler in some ways, compared to the case of wave equations, it is also more degenerate in the following sense: the regularity theory for the eikonal function is less favorable in that is one degree less differentiable in some directions compared to the case of wave equations. We therefore must close the proof of Theorem 1.5 under this decreased differentiability. We defer further discussion of this point until Subsubsect. 1.6.5. Here, we will simply further motivate our use of nonlinear geometric optics in proving shock formation.

    First, we note that in more than one spatial dimension, it does not seem possible to close the proof using only the Cartesian coordinates; indeed, Theorem 1.5 shows that the blowup of precisely corresponds to the vanishing of the inverse foliation density of the characteristics, which is equivalent to the blowup of . Hence, it is difficult to imagine how a sharp, constructive proof of stable blowup would work without referencing an eikonal function. In view of these considerations, we construct a geometric coordinate system adapted to the transport operator vectorfield and prove that , , , and their geometric coordinate partial derivatives remain regular all the way up to the singularity in . The blowup of occurs because the change of variables map between geometric and Cartesian coordinates degenerates, which is in turn tied to the vanishing of ; the Jacobian determinant of this map is in fact proportional to ; see Lemma 3.25. The coordinate is the standard Cartesian time function. The geometric coordinate function is the eikonal function described in Theorem 1.5. The initial condition is adapted to the approximate plane symmetry of the initial data. We similarly construct the “geometric torus coordinates” by solving with the initial condition . The main challenge is to derive regular estimates relative to the geometric coordinates for all quantities, including the solution variables and quantities constructed out of the geometric coordinates.

  • (Full quasilinear coupling) Because we are able to close the proof with decreased regularity for (compared to the case of wave equations), we are able to handle full quasilinear coupling between all solution variables. This is an interesting advancement over prior works, where the principal coefficients in the evolution equation for the shock-forming variable were allowed to depend only on the shock-forming variable itself and on other solution variables that satisfy a wave equation with the same principal part as the shock-forming variable; i.e., in equation (1.3.2), we allow , where the principal part of the evolution equation (1.3.3) for is distinct (by assumption) from .

1.6. A more detailed overview of the proof

In this subsection, we provide an overview of the proof of our main results. Our analysis is based in part on some key ideas originating in earlier works, which we review in Subsect. 1.7. Our discussion in this subsection is, at times, somewhat loose; our rigorous analysis begins in Sect. 2.

1.6.1. Setup and geometric constructions

In Sects. 2-3, we construct the geometric coordinate system described in Subsect. 1.2, which is central for all that follows. We also construct many related geometric objects, including the inverse foliation density (see Def. 3.5 for the precise definition) of the characteristics of the eikonal function . As we mentioned earlier, our overall strategy is to show that the solution remains regular with respect to the geometric coordinates, all the way up to the top derivative level, to show that vanishes in finite time, and to show that the vanishing of is exactly tied to the blowup of . It turns out that when deriving estimates, it is important to replace the geometric coordinate partial derivative vectorfield with a -tangent vectorfield that we denote by , which is similar to but generally not parallel to it; see Fig. 1 for a picture of . In the context of the present paper, the main advantage of is that it enjoys the following key property: the vectorfield has Cartesian components that remain uniformly bounded, all the way up to the shock. Put differently, we have , where we will show that is a vectorfield of order-unity Euclidean length and thus the Euclidean length of is . We further explain the significance of this in Subsubsect. 1.6.4, when we outline the proof that the shock forms. In total, when deriving estimates for the derivatives of quantities, we differentiate them with respect to elements of the vectorfield frame

(1.6.1)

which spans the tangent space at each point with . Here, , (where the partial differentiation is with respect to the geometric coordinates), is the vectorfield from (1.3.2) and, by construction, we have (see (3.3.5)). The vectorfields and are tangent to the , while is transversal and normalized by (see (3.3.6)); see Fig. 1 on pg. 1 for a picture of the frame. Note that since is of length , the uniform boundedness of is consistent with the formation of a singularity in the Cartesian coordinate partial derivatives of when ; see Subsubsect. 1.6.4 for further discussion of this point.

We now highlight a crucial ingredient in our proof: we treat the Cartesian coordinate partial derivatives of as independent unknowns , defined by

(1.6.2)

As we stressed already in Subsect. 1.2, our reliance on allows us to avoid commuting equation (1.3.3) up to top order with elements of , which allows us to avoid certain top-order commutator terms that would result in the loss of a derivative. Moreover, as we noted in Theorem 1.5, a key aspect of our framework is to show that the quantities remain bounded up to the singularity in . To achieve this, we will control by studying its evolution equation , whose inhomogeneous terms are controllable under the scope of our approach.

1.6.2. A more precise description of 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 of thickness and on a portion of the characteristic , where is a parameter, fixed until Theorem 10.1; see Fig. 2 on pg. 2. We will study spacetime regions such that , where is the eikonal function described above. We have introduced the parameter because one would need to allow to vary in order to study the behavior of the solution up the boundary of the maximal development, as we mentioned in Remark 1.8.

In our analysis, we will use a bootstrap argument in which we only consider times with , where is a data-dependent parameter described in Subsubsect. 1.6.3 (see also Def. 5.1). Our main theorem shows that if , then a shock forms at a time equal to a small perturbation of ; see Subsubsect. 1.6.4 for an outline of the proof. For this reason, in proving our main results, we only take into account only the portion of the data lying in and in the subset of the characteristic ; from domain of dependence considerations, one can infer that only this portion can influence the solution in the regions under study.

Remark 1.9.

For the remainder of Subsect. 1.6, we will suppress further discussion of by setting .

1.6.3. Data-size assumptions, bootstrap assumptions, and pointwise estimates

In Sect. 5, we state our assumptions on the data and formulate bootstrap assumptions that are useful for deriving estimates. Our assumptions on the data involve the parameters , , , and , where, for our proofs to close, must be chosen to be small in an absolute sense and must be chosen to be small in a relative sense compared to and (see Subsect. 5.4 for a precise description of the required smallness). The following remarks capture the main ideas behind the data-size parameters.

  1. is the size of .

  2. is the size, in appropriate norms, of the derivatives of up to top order in which at least one -tangential differentiation occurs, and of , and all of their derivatives up to top order with respect to elements of from (1.6.1). We emphasize that we will study perturbations of plane symmetric shock-forming solutions such that . That is, the case corresponds to a plane symmetric simple wave in which . We state the total number of derivatives that we use to close the estimates in Subsect. 5.1 and Subsubsect. 5.2.2. We also highlight that to close our proof, we never need to differentiate any quantity with more than one copy of the -transversal vectorfield . This is possible in part because of the following crucial fact, proved in Lemma 3.22: commuting the elements of the frame with each other yields a vectorfield belonging to .

  3. is the size of the -transversal derivative of .

  4. , is a modified measure of the size of the -transversal derivative of , where is a coefficient determined by the nonlinearities and .

  5. When , other geometric quantities that we use in studying solutions obey similar size estimates, where any differentiation of a quantity with respect to a -tangential vectorfield leads to -smallness; see Lemma 5.4. A crucial exception occurs for , which initially is of relatively large size in view of its evolution equation (see (3.7.1a) for the precise evolution equation).

  6. The relative smallness of corresponds to initial data that are close to that of a simple plane symmetric wave, as we described in Subsect. 1.4.

One of the main steps in our analysis is to propagate the above size assumptions all the way up to the shock. To this end, on a region of the form , we make -type bootstrap assumptions that capture the expectation that the above size assumptions hold. In particular, the bootstrap assumptions capture our expectation that no singularity will form in any quantity relative to the geometric coordinates. Moreover, since , the bootstrap assumptions for the smallness141414We note that the bootstrap assumptions refer to a parameter that, in our main theorem, we will show is controlled by ; for brevity, we will avoid further discussion of until Subsubsect. 5.3.2. of capture our expectation that the Cartesian coordinate partial derivatives of should remain bounded; indeed, this is a key aspect of our proof that we use to control various error terms depending on . As we mentioned earlier, a crucial point is that we have set the problem up so that the shock forms at time . Therefore, we make the assumption

(1.6.3)

which leaves us with ample margin of error to show that a shock forms. In particular, in view of (1.6.3), we can bound factors of , , etc. by a constant depending on , and the estimates will close as long as is sufficiently small; see Subsect. 1.8 for further discussion on our conventions regarding the dependence of constants .

In Sect. 6, with the help of the bootstrap assumptions and data-size assumptions described above, we commute all evolution equations, including (1.3.2)-(1.3.3) and evolution equations for and related geometric quantities, with elements of the up to top order and derive pointwise estimates for the error terms. Actually, due to the special structures of the equations relative to the geometric coordinates, we never need to commute the evolution equations verified by , , or with the transversal vectorfield . Moreover, for the other geometric quantities, we need to commute their evolution equations at most once with . We clarify, however, that we commute all equations many times with the elements of the -tangential subset .

1.6.4. Sketch of the formation of the shock

Let us assume that the bootstrap assumptions and pointwise estimates described in Subsubsect. 1.6.3 hold for a sufficiently long amount of time. We will sketch how they can be used to give a simple proof of shock formation, that is, that and blows up. The main estimates in this regard are provided by Lemma 6.8; here we sketch them. First, using equation (3.7.1a), the bootstrap assumptions, and the pointwise estimates, we deduce the following evolution equation for the inverse foliation density: , where the “blowup coefficient” was described in Subsubsect. 1.6.3 and denotes small error terms, which we ignore here. Next, we note the following pointwise estimate, which falls under the scope of the discussion in Subsubsect. 1.6.3: (smallness is gained since is a -tangential differentiation). Recalling that , we use the fundamental theorem of calculus to deduce . Inserting this estimate into the above one for , we obtain . From the fundamental theorem of calculus and the initial condition , we obtain . From this estimate and the definition of , we obtain . Hence, vanishes for the first time at , as desired. Moreover, the above reasoning can easily be extended to show that at any point such that . Recalling that where has order-unity Euclidean length, we see the following:

must blow up like as .

This argument shows, in particular, that the vanishing of exactly coincides with the blowup of .

1.6.5. Considerations of regularity

This subsubsection is an interlude in which we highlight some issues tied to considerations of regularity. Our discussion will distinguish the problem of shock formation for transport equations from the (by now) well-understood case of wave equations, which we further describe in Subsubsect. 1.7.2. To illustrate the issues, we will highlight some features of our analysis, with a focus on derivative counts. In Lemma 3.21, we derive the following evolution equation for the Cartesian components of : , where . Recalling that , that , and that is a smooth function of , we infer, from standard energy estimates for transport equations, that should have the same degree of Sobolev differentiability as and . In particular, we expect that should be one degree less differentiable than . For similar reasons, , , and some other geometric quantities that play a role in our analysis are also one degree less differentiable than . The following point is crucial for our approach:

We are able to close the energy estimates for up to top order even though, upon commuting ’s transport equation, we generate error terms that depend on the “less differentiable” quantities.

That is, in controlling , we must carefully ensure that all error terms feature an allowable amount of regularity. Moreover, the same careful care must be taken throughout the paper, by which we mean that we must ensure that we can close the estimates for all quantities using a consistent number of derivatives. In particular, we stress that it is precisely due to considerations of the regularity of the Cartesian components of and that we have introduced the quantities , as we explained in Subsubsect. 1.6.1.

In the case of wave equations, the derivative counts are different. For example, the inverse foliation density enjoys the same Sobolev regularity as the wave equation solution variable in directions tangent to the characteristics, a gain of one tangential derivative compared to the present work. For wave equations, a similar gain in tangential differentiability also holds for some other key geometric objects, which we will not describe here. The gain is available because certain special combinations of quantities constructed out of the eikonal equation and the wave equation solution variable satisfy an unexpectedly good evolution equation, with source terms that have better than expected regularity; see Subsubsect. 1.7.2 or the survey article [gHsKjSwW2016] for further discussion. Moreover, this gain seems essential for closing some of the top-order energy estimates in the wave equation case, the reason being that one must commute the geometric vectorfields through the second-order wave operator, which eats up the gain. As we explain in Subsubsect. 1.7.2, one pays a steep price in gaining back the derivative: the resulting energy estimates allow for possible energy blowup at the high derivative levels, a difficulty which we do not encounter in the present work.

We close this subsubsection by again highlighting that we are able to handle systems with full quasilinear coupling (in the sense explained in the second paragraph of Subsect. 1.2) precisely because we are able to close our estimates using geometric quantities that are one degree less differentiable than . In contrast, the special combinations of quantities mentioned in the previous paragraph, which are needed to close the wave equation energy estimates, seem to be unstable under a full quasilinear coupling of multiple speed wave systems. Here is one representative manifestation of this issue: the problem of multi-space-dimensional shock formation for covariant wave equation systems (see Footnote 20 on pg. 20 regarding the notation) of the form

is open whenever , even though shock formation for systems with and for scalar equations is well-understood.

1.6.6. Energy estimates

In Sect. 8, we derive the main technical estimates of the article: energy estimates up to top order for , , , , and related geometric quantities. Energy estimates are an essential ingredient in the basic regularity theory of quasilinear hyperbolic systems in multiple spatial dimensions, and in this article, they are also important because they yield improvements of our bootstrap assumptions described in Subsubsect. 1.6.3. We now describe the energies, which we construct in Sect. 4. To control the transport variable , we construct geometric energies along . To control the symmetric hyperbolic variables and , we construct -weighted energies along as well as non--weighted energies along the characteristics . With defined to be the subset of in which the eikonal function takes on values in between and and defined to be the subset of corresponding to times in between and , we have, with ,

(1.6.4a)
(1.6.4b)
(1.6.4c)

In our analysis, we of course must also control various higher-order energies, but here we ignore this issue. The degenerate weights featured in and arise from expressing the standard energy for symmetric hyperbolic systems in terms of the geometric coordinates. For controlling certain error integrals that arise in the energy identities, it is crucial that the characteristic fluxes and do not feature any degenerate weight. These characteristic fluxes are positive definite only because our structural assumptions on the equations ensure that the propagation speed of and is strictly slower than that of (see (2.3.1) for the precise assumptions). Readers can consult Lemma 4.2 and its proof to better understand the role of these assumptions.

We now outline the derivation of the energy estimates; see Sect. 8 for precise statements and proofs. Let us define151515Our definition of given here is schematic. See Def. 8.1 for the precise definition of the controlling quantity, which we denote by . the controlling quantity to be the sum of the terms on LHSs (1.6.4a)-(1.6.4c) and their analogs up to the top derivative level (corresponding to differentiations with respect to the geometric vectorfields). The initial data that we treat are such that and , with the small parameter described in Subsubsect. 1.6.3. We again stress that for simple plane waves. Energy identities, based on applying the divergence theorem on the geometric coordinate region , together with the pointwise estimates for error terms mentioned in Subsubsect. 1.6.3, lead to the following inequality:

(1.6.5)

where the terms depend on other geometric quantities and can be bounded using similar arguments similar to the ones we sketch here. In view of the definition of , we deduce the following inequality from (1.6.5):

(1.6.6)

Then from (1.6.6) and Gronwall’s inequality with respect to and , we conclude, ignoring the terms and taking into account (1.6.3), that the following a priori estimate holds for (see Prop. 8.6 for the details):

(1.6.7)

The estimate (1.6.7) represents the realization of our hope that the solution remains regular relative to the geometric coordinates, up to the top derivative level.

We now stress the following key point: the characteristic fluxes and are needed to control the terms on RHS (1.6.5); without the characteristic fluxes, instead of the term on RHS (1.6.6), we would instead have the term , whose denominator vanishes as the shock forms. Such a term would have led to a priori estimates allowing for the possibility that at all derivative levels, the geometric energies blow up as the shock forms. This in turn would have been inconsistent with the bootstrap assumptions described in Subsubsect. 1.6.3 and would have obstructed our goal of showing that the solution remains regular relative to the geometric coordinates.

1.6.7. Combining the estimates

Once we have obtained the a priori energy estimates, we can derive improvements of our -type bootstrap assumptions via Sobolev embedding (see Cor. 8.8). These steps, together with the estimates from Subsubsect. 1.6.4 showing that vanishes in finite time, are the main steps in the proof of the main theorem. We need a few additional technical results to complete the proof, including some results guaranteeing that the geometric and Cartesian coordinates are diffeomorphic up to the shock (see Sect. 7) and some fairly standard continuation criteria (see Sect. 9), which in total ensure that the solution survives up to the shock. We combine all of these results in Sect. 10, where we prove the main theorem.

1.7. Connections to prior work

Many aspects of the approach outlined in Subsect. 1.6 have their genesis in earlier works, which we now describe.

1.7.1. Results in one spatial dimension

In one spatial dimension and in symmetry classes whose PDEs are effectively one-dimensional, there are many results, by now considered classical, that use the method of characteristics to exhibit the formation of shocks in initially smooth solutions to various quasilinear hyperbolic systems. Important examples include Riemann’s work [bR1860] (in which he developed the method of Riemann invariants), Lax’s proof [pL1964] of stable blowup for genuinely nonlinear systems via the method of Riemann invariants, Lax’s blowup results [pL1972, pL1973] for scalar conservation laws, John’s extension [fJ1974] of Lax’s work to systems in one spatial dimension with more than two unknowns (which required the development of new ideas since the method of Riemann invariants does not apply), and the recent work [dCdRP2016] of Christodoulou–Raoul Perez, in which they significantly sharpened John’s work [fJ1974]. The main obstacle to extending the above results to more than one spatial dimension is that one must complement the method of characteristics with an ingredient that, due to the singularity formation, is often accompanied by enormous technical complications: energy estimates that are adapted to and that hold up to the singularity. We further explain these technical complications in the next subsubsection.

1.7.2. Results in more than one spatial dimension

The first breakthrough results on shock formation in more than one spatial dimension without symmetry assumptions were proved by Alinhac [sA1999a, sA1999b, sA2001b] for small-data solutions to scalar quasilinear wave equations of the form

(1.7.1)

that fail to satisfy the null condition. Here, is a Lorentzian metric161616That is, the matrix of Cartesian components of has signature . equal to the Minkowski metric plus an error term of size . As we do in this paper, Alinhac constructed a set of geometric coordinates tied to an eikonal function , which in the context of his problems was a solution the fully nonlinear eikonal equation

(1.7.2)

Much like in our work here, the level sets of are characteristic hypersurfaces for equation (1.7.1). They are also known as null hypersurfaces in the setting of Lorentzian geometry in view of their intimate connection to the -null171717That is, if , then by (1.7.2), we have . vectorfield . In his works, Alinhac identified a set of small compactly supported initial data verifying a non-degeneracy condition such that blows up in finite time due to the intersection of the characteristics while and remain bounded. Moreover, relative to the geometric coordinates, and remain smooth, except possibly at the very high derivative levels (we will elaborate upon this just below).

In proving his results, Alinhac faced three serious difficulties. We will focus only on the case of three spatial dimensions though Alinhac obtained similar results in two spatial dimensions. The first difficulty is that for small data, solutions to (1.7.1) experience a long period of dispersive decay, which seems to work against the formation of a shock and which necessitated the application of Klainerman’s commuting vectorfield method [sK1985, sK1986] in which the vectorfields have time and radial weights. We stress that such dispersive behavior is not exhibited by the solutions that we study in this article and hence our vectorfields do not feature time or radial weights. Alinhac showed that after an era181818Roughly for a time interval of length , with the size of the data in a weighted Sobolev norm. of dispersive decay, the nonlinearity in equation (1.7.1) takes over and drives the formation of the shock. The second main difficulty faced by Alinhac is that to follow the solution up the singularity, it seems necessary to commute the equations with geometric vectorfields constructed out of the eikonal function, and these vectorfields seem to lead to the loss of a derivative when commuted through the wave operator. Specifically, the geometric vectorfields have Cartesian components that depend on , and hence commuting them through the wave equation (1.7.2) leads to an equation of the schematic form . The difficulty is that standard wave equation energy estimates suggest that, due to the source term , enjoys only the same Sobolev regularity as , whereas standard energy estimates for the eikonal equation (1.7.2) only allow one to prove that enjoys the same Sobolev regularity as ; this suggests that the approach of using vectorfields constructed out of an eikonal function will lead to the loss of a derivative. To overcome this difficulty, Alinhac obtained the nonlinear solution, up to the shock, as the limit of iterates that solve singular linearized problems, and he used a rather technical Nash–Moser iteration scheme featuring a free boundary in order to recover the loss of a derivative. For technical reasons, his reliance on the Nash–Moser iteration allowed him to follow “most” small-data solutions to the constant-time hypersurface of first blowup, and not further. More precisely, his approach only allowed him to treat “non-degenerate” data such that the first singularity is isolated in the constant-time hypersurface of first blowup. We stress that in our work here, we encounter a similar difficulty concerning the regularity of the geometric vectorfields, but since our PDE systems are first-order, we are able to overcome it in a different way, without relying on a Nash–Moser iteration scheme; see Subsect. 1.2 and Subsubsect. 1.6.5. The third and most challenging difficulty encountered by Alinhac is the following: when proving energy estimates relative to the geometric coordinates, it seems necessary to rely on energies that feature degenerate weights that vanish as the shock forms; the weights are direct analogs of the inverse foliation density from Theorem 1.5. These weights make it difficult to control certain error terms in the energy identities, which in turn leads to a priori estimates allowing for the following possibility: as the shock forms, the high-order energies might blow up at a rate tied to . We stress that the possible high-order energy blowup encountered by Alinhac occurs relative to the geometric coordinates and is distinct from the formation of the shock singularity (in which blows up). To close the proof, Alinhac had to show that the possible high-order geometric energy blowup does not propagate down too far to the lower geometric derivative levels, i.e., that the solution remains smooth relative to the geometric coordinates at the lower derivative levels. This “descent scheme” costs many derivatives, and for this reason, the data must belong to a Sobolev space of rather high order for the estimates to close. We stress that although the energies that we use in the present paper also contain the same degenerate weights, we encounter different kinds of error terms in our energy estimates, tied in part to the fact that our systems are first-order and tied in part to our strategy of estimating the quantity defined by (1.6.2). For this reason, our a priori estimates energy relative to the geometric coordinates are regular in that even the top-order geometric energies remain uniformly bounded up to the shock.

In Christodoulou’s remarkable work [dC2007], he significantly sharpened Alinhac’s shock formation results for the quasilinear wave equations of irrotational (i.e., vorticity-free) relativistic fluid mechanics in three spatial dimensions, which form a sub-class of wave equations of type (1.7.1). These wave equations arise from formulating the relativistic Euler equations in terms of a fluid potential , which is possible when the vorticity vanishes. The equations studied by Christodoulou enjoy special features that he exploited in his proofs, such as having an Euler-Lagrange formulation with a Lagrangian that is invariant under the Poincaré group. The main results proved by Christodoulou are as follows: i) there is an open (relative to a Sobolev space of high, non-explicit order) set of small191919In the context of [dC2007], “small” means a small perturbation of the non-trivial constant-state solutions, which take the form , where is a constant. data such that the only possible singularities that can form in the solution are shocks driven by the intersection of the characteristics; ii) there is an open subset of the data from i), not restricted by non-degeneracy assumptions of the type imposed by Alinhac, such that a shock does in fact form in finite time; and iii) for those solutions that form shocks, Christodoulou gave a complete description of the maximal classical development of the data near the singularity, which intersects the future of the constant-time hypersurface of first blowup. His sharp description of the maximal development seems necessary for even properly setting up the shock development problem. This is the problem of uniquely locally continuing the solution past the singularity to the Euler equations in a weak sense, a setting in which one must also construct the “shock hypersurface,” across which the solution jumps (being smooth on either side of it). The shock development problem in relativistic fluid mechanics was solved in spherical symmetry by Christodoulou–Lisibach in [dCaL2016] and, by Christodoulou in yet another breakthrough work [dC2017], for the non-relativistic compressible Euler equations without symmetry assumptions in a restricted case (known as the restricted shock development problem) such that the jump in entropy across the shock hypersurface is ignored.

Compared to Alinhac’s approach, the main technical improvement afforded by Christodoulou’s approach [dC2007] to proving shock formation is that it avoids the loss of a derivative through a sharper, more direct method; instead of using Alinhac’s Nash–Moser scheme, Christodoulou found special combinations of geometric quantities that satisfy good evolution equations, and he combined them with elliptic estimates on co-dimension two spacelike hypersurfaces. This approach to avoiding the loss of a derivative in wave equation eikonal functions originated in the aforementioned proof [dCsK1993] of the stability of Minkowski spacetime, and it was extended by Klainerman–Rodnianski [sKiR2003] to the case of general scalar quasilinear wave equations in their study of low-regularity well-posedness for wave equations of the form . In total, this allowed Christodoulou to control the solution up to the shock using a traditional “forwards” approach, without the free boundary found in Alinhac’s iteration scheme. However, as in Alinhac’s work, Christodoulou’ energy estimates allowed for the possibility that the high-order energies might blow up. Christodoulou therefore had to give a separate, technical argument to show that any high-order energy singularity does not propagate down too far to the lower geometric derivative levels.

In [jS2016b], we extended Christodoulou’s sharp shock formation results to the case of general quasilinear wave equations of type (1.7.1) in three spatial dimensions that fail to satisfy the null condition, to the case of covariant wave equations of the type202020Here, is the covariant wave operator of . Relative to arbitrary coordinates, . that fail to satisfy the null condition, and to inhomogeneous versions of these wave equations featuring “admissible” semilinear terms. Similar results were proved in [dCsM2014] for a subset of these equations, namely those wave equations arising from non-relativistic compressible fluid mechanics with vanishing vorticity. All of the results mentioned so far in this subsubsection are explained in detail in the survey article [gHsKjSwW2016].

In the wake of the above results, there have been significant further advancements, which we now describe. In [jSgHjLwW2016], we extended the shock formation results of [jS2016b] to a new, physically relevant regime of initial conditions in two spatial dimensions such that the solutions are close to simple outgoing plane symmetric waves, much like the setup of the present article. For the initial conditions studied in [jSgHjLwW2016], the solutions do not experience dispersive decay. Hence, we used a new analytic framework to control the solution up to the shock, based on “close-to-simple-plane-wave”-type smallness assumptions on the data that are similar in spirit to the assumptions that we make on the data in the present article. For special classes of wave equations in three spatial dimensions with cubic nonlinearities, Miao–Yu [sMpY2017] proved similar shock formation results for a class of large initial data featuring a single scaling parameter, similar to the short pulse ansatz exploited by Christodoulou in his breakthrough work [dC2009] on the formation of trapped surfaces in solutions to the Einstein-vacuum equations. For the same wave equations studied in [sMpY2017], Miao [sM2016] recently used a related but distinct ansatz for the initial data to prove the existence of an open set of solutions that blow up at time but exist classically on the time interval .

All of the above works concern systems that feature relatively simple characteristics: those corresponding to a single wave operator. We now describe some recent shock formation results in which the systems have more complicated principal parts, leading to multiple speeds of propagation and distinct families of characteristics. The first result of this type without symmetry assumptions was our joint work [jLjS2016b] with J. Luk, which concerned the compressible Euler equations in two spatial dimensions under an arbitrary212121There is one exceptional equation of state, known as that of the Chaplygin gas, to which the results of [jLjS2016b] do not apply. In one spatial dimension, the resulting PDE system is totally linearly degenerate, and many experts believe that shocks do not form in solutions to such systems. barotropic222222A barotropic equation of state is such that the pressure is a function of the density. equation of state. Specifically, in [jLjS2016b], we extended the shock formation results of [dCsM2014] for the compressible Euler equations to allow for the presence of small amounts of vorticity at the location of the singularity. The vorticity satisfies a transport equation and, as it turns out, remains Lipschitz with respect to the Cartesian coordinates, all the way up to the shock. More precisely, the shock occurs in the “sound wave part” of the system rather than in the vorticity, and, as in all prior works, it is driven by the intersection of a family of characteristic hypersurfaces corresponding to a Lorentzian metric (known as the acoustical metric in the context of fluid mechanics). In particular, [jLjS2016b] yielded the first proof of stable shock formation without symmetry assumptions in solutions to a hyperbolic system featuring multiple speeds, where all solution variables were allowed to interact up to the singularity.

The results proved in [jLjS2016b] were based on a new wave-transport-div-curl formulation of the compressible Euler equations under a barotropic equation of state, which we derived in [jLjS2016a]. The new formulation exhibits remarkable null structures and regularity properties, tied in part to the availability of elliptic estimates for the vorticity in three spatial dimensions (vorticity stretching does not occur in two spatial dimensions, and in its absence, one does not need elliptic estimates to control the vorticity). In a forthcoming work, we will extend the shock formation results of [jLjS2016b] to the much more difficult case of three spatial dimensions, where to control the vorticity up to top order in a manner compatible with the wave part of the system, one must rely on the elliptic estimates, which allow one to show that the vorticity is exactly as differentiable as the velocity with respect to geometric vectorfields adapted to the sound wave characteristics. In [jS2017a], we extended the results of [jLjS2016a] to allow for an arbitrary equation of state in which the pressure depends on the density and entropy. The formulation of the equations in [jS2017a] exhibits further remarkable properties that, in our forthcoming work, we will use to prove a stable shock formation result in three spatial dimensions in which the vorticity and entropy are allowed to be non-zero at the singularity. In [jS2017b], in two spatial dimensions, we proved the first stable shock formation result for systems of quasilinear wave equations featuring multiple wave speeds of propagation, i.e., the systems featured more than one distinct quasilinear wave operator. The main result provided an open set of data such that the “fastest” wave forms a shock in finite time while the remaining solution variables remain regular up to the singularity in the fast wave, much like in Theorem 1.5. The initial conditions were perturbations of simple plane waves, similar to the setup for the case of the scalar wave equations studied in [jSgHjLwW2016] and similar to the setup of the present article. The main new difficulty that we faced in [jS2017b] is that the geometric vectorfields adapted to the shock-forming fast wave, which seem to be an essential ingredient for following the fast wave all the way to its singularity, exhibit very poor commutation properties with the slow wave operator. Indeed, commuting the geometric vectorfields all the way through the slow wave operator produces error terms that are uncontrollable both from the point of view of regularity and from the point of view of the strength of the singular commutator terms that this generates. To overcome this difficulty, we relied on a first-order reformulation of the slow wave equation which, though somewhat limiting in the precision it affords, allows us to avoid commuting all the way through the slow wave operator and hence to avoid the uncontrollable error terms.

1.8. Notation, index conventions, and conventions for “constants”

We now summarize some our notation. Some of the concepts referred to here are defined later in the article. Throughout, denote the standard Cartesian coordinates on spacetime , where is the time variable and are the space variables. We denote the corresponding Cartesian partial derivative vectorfields by (which are globally defined and smooth even though are only locally defined) and we often use the alternate notation and .

  • Lowercase Greek spacetime indices , , etc. correspond to the Cartesian spacetime coordinates and vary over . Lowercase Latin spatial indices ,, etc. correspond to the Cartesian spatial coordinates and vary over . An exception to the latter rule occurs for the geometric torus coordinate vectorfields from (3.1.5), in which the labeling index varies over . Uppercase Latin indices such as correspond to the components of the array of symmetric hyperbolic variables and typically vary from to .

  • We use Einstein’s summation convention in that repeated indices are summed over their respective ranges.

  • Unless otherwise indicated, all quantities in our estimates that are not explicitly under an integral are viewed as functions of the geometric coordinates of Def. 3.4. Unless otherwise indicated, quantities under integrals have the functional dependence established below in Def. 3.26.

  • If and are two operators, then denotes their commutator.

  • means that there exists such that .

  • means that and .

  • means that .

  • Constants such as and 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 and from Subsect. 5.2. However, the constants can be chosen to be independent of the parameters , , and whenever the following conditions hold: i) and are sufficiently small relative to , relative to , and relative to , and ii) is sufficiently small relative to in the sense described in Subsect. 5.4.

  • Constants are universal in that, as long as and are sufficiently small relative to , they do not depend on , , , or .

  • means that , with as above.

  • and respectively denote the standard floor and ceiling functions.

2. Rigorous setup of the problem and fundamental definitions

In this section, we state the equations that we will study and state our basic assumptions on the nonlinearities.

2.1. Statement of the equations

Our main results concern systems in spacetime dimensions and unknowns of the following form:

(2.1.1)
(2.1.2)

where the scalar function will eventually form a shock, is an integer,232323Our results also apply in the case , though we omit discussion of this simpler case.

(2.1.3)

denotes the “symmetric hyperbolic variables” (which will remain regular up to the singularity in ), is a vectorfield whose Cartesian components are given smooth functions of and , that is, , and are symmetric matrices whose components are given smooth functions of and . Note that equation (2.1.2) is equivalent to the scalar equations , where and with summation over and . For convenience, we assume the normalization conditions

(2.1.4a)
(2.1.4b)

More generally, if , then (2.1.4a)-(2.1.4b) can be achieved by performing a linear change of coordinates in the plane and then dividing equation (2.1.1) by a scalar.

As we stressed in the introduction, an essential aspect of our analysis is that we treat the Cartesian coordinate partial derivatives of as independent quantities. For this reason, we define

(2.1.5)

As a straightforward consequence of equation (2.1.2) and definition (2.1.5), we obtain the following evolution equation for :

(2.1.6)

2.2. The genuinely nonlinear-type assumption

To ensure that shocks can form in nearly plane symmetric solutions, we assume that for sufficiently small, we have

(2.2.1)

2.3. Assumptions on the speed of propagation for the symmetric hyperbolic subsystem

In this subsection, we state our assumptions on the speed of propagation for the symmetric hyperbolic subsystem (2.1.2). Specifically, we assume that the matrices

(2.3.1)

We now explain the significance of (2.3.1). The positivity of ensures that for solution values near the “background state” , the hypersurfaces are spacelike for equation (2.1.2), that is, for the evolution equation verified by the non-shock-forming variable . By (2.1.4a), the are also spacelike for equation (2.1.1), i.e., is transversal to . The positivity of will ensure that for solution values near the background state, hypersurfaces close to the flat planes are spacelike for equation (2.1.2). This assumption is significant because for the solutions that we will study, we will construct (in Subsect. 3.1) a family of hypersurfaces that are characteristic for equation (2.1.1) (that is, for the operator ) and that are close to the flat planes . Put differently, the will be characteristic for the evolution equation for but spacelike for the evolution equation for , which essentially means that for solution values near the background state, propagates at a strictly faster speed than (and also strictly faster than , since the principal coefficients in the evolution equations for and are the same).

3. Geometric constructions

In this section, we define/construct most of the geometric objects that we use to analyze solutions. We defer the construction of the energies until Sect. 4.

3.1. The eikonal function and the geometric coordinates

In this subsection, we construct the geometric coordinates that we use to follow the solution all the way to the shock. The most important of these is the eikonal function.

Definition 3.1 (Eikonal function).

The eikonal function is the solution to the following transport initial value problem, where is the transport operator vectorfield from equation (2.1.1):

(3.1.1)

For reasons described in Remark 1.8 and Subsubsect. 1.6.2, we now fix a real parameter verifying

(3.1.2)

We will restrict out attention to spacetime regions with .

Our analysis will take place on the following subsets of spacetime, which are tied to the eikonal function; see Fig. 2 for a picture of the setup.

Definition 3.2 (Subsets of spacetime).

We define the following subsets of spacetime:

(3.1.3a)
(3.1.3b)
(3.1.3c)
(3.1.3d)