arXiv subjects

Jared Speck

Publications and source records attributed to Jared Speck.

At least 19 recordsLinked to original sources

arXiv2026

Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime

We prove a pair of complementary small-data stability theorems describing the global future and past structure of classical solutions to the isentropic, spherically symmetric $3D$ compressible Euler equations. We allow any equation of state with positive sound speed, except that our shock-formation results do not apply to the Chaplygin gas. Our first theorem concerns open sets of smooth Cauchy data that are perturbations of trivial data with vanishing velocity and constant positive density. The perturbed data have finite kinetic energy, and the perturbed density is equal to a positive constant plus a signed ``asymptotically flat'' tail. Our main theorem yields a complete description of the existence and uniqueness of the maximal globally hyperbolic development (MGHD) of the data. The boundary of the MGHD contains hypersurfaces that extend to spatial infinity on which our solutions develop gradient-singularities. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-$D$ quasilinear wave system. While prior works have yielded local portions of ``candidate'' MGHDs, neither the existence nor the uniqueness of an MGHD can be inferred from local considerations. Our second theorem considers analogous data with the opposite sign of the $1/r$ density tail. We prove global existence throughout spacetime, to both the future and past. The solution's asymptotic behavior towards null infinity is not linear, but rather is distorted by a logarithmic bending of the sound cones away from the flat Minkowski ones, leading to logarithmically enhanced dispersion. This is the first small-data global existence result in the entire spacetime for any $3D$ quasilinear wave system that fails to satisfy the null condition and the weak null condition. The main mechanism of stabilization is a global-in-spacetime rarefaction effect, tied to the $1/r$ tail.

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arXiv2026

The emergence of the Cauchy horizon from the crease for $3D$ compressible Euler flow

We study open sets of nearly plane symmetric data for the $3D$ compressible Euler equations with dynamic entropy and non-trivial vorticity. In our prior work, we proved that the solutions develop a gradient singularity along a singular boundary, which is a hypersurface that emanates from a co-dimension 2, spacelike submanifold called the crease. In the present paper, we prove that a null hypersurface, called a Cauchy horizon, emanates from the crease, and propagates in a direction transverse to the singular boundary. The solution remains smooth up to the Cauchy horizon, even though it ``feels'' the influence of the gradient singularity at the crease. The union of the Cauchy horizon, the crease, and the singular boundary, make up a connected portion of the boundary of a maximal globally hyperbolic development (MGHD) of the data. Roughly, an MGHD is the ``largest'' way that smooth data can evolve into a classical solution. Known examples show that the only way one can guarantee uniqueness of an MGHD is by proving that certain structural properties hold along its entire boundary. We prove that a local version of the needed properties are satisfied along the Cauchy horizon and singular boundary. Our work provides the first description of the formation, structure, and stability of an $O(1)$-size portion of the Cauchy horizon in $3D$ without symmetry assumptions. The presence of dynamic entropy and vorticity stretching, both of which are absent in simpler settings such as $2D$ isentropic Euler, introduces significant difficulties that we resolve with novel techniques. Our approach relies on new foliations of spacetime dynamically adapted to the shape of the crease, and a PDE framework based on double-null foliations and the energy identities we derived in earlier work. Collectively, our techniques allow us to handle a challenging new solution regime where the crease lacks strict convexity.

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arXiv2025

A quasilinear wave with a supersonic shock in a weak solution interrupting the classical development

We study the Cauchy problem for classical and weak shock-forming solutions to a model quasilinear wave equation in $1+1$ dimensions arising from a convenient choice of $C^{\infty}$ initial data, which allows us to solve the equation using elementary arguments. The simplicity of our model allows us to succinctly illustrate various phenomena of geometric and analytic significance tied to shocks, which we view as a prototype for phenomena that can occur in more general quasilinear hyperbolic PDE solutions. Our Cauchy problem admits a classical solution that blows up in finite time. The classical solution is defined in a largest possible globally hyperbolic region called a maximal globally hyperbolic development (MGHD), and its properties are tied to the intrinsic Lorentzian geometry of the equation and solution. The boundary of the MGHD contains an initial singularity, a singular boundary along which the solution's second derivatives blow up (the solution and its first derivatives remain bounded), and a Cauchy horizon. Our main results provide the first example of a provably unique MGHD for a shock-forming quasilinear wave equation solution; it is provably unique because its boundary has a favorable global structure that we precisely describe. We also prove that for the same $C^{\infty}$ initial data, the Cauchy problem admits a second kind of solution: a unique global weak entropy solution that has a shock curve separating two smooth regions. Of particular interest is our proof that the classical and weak solutions agree before the shock but differ in a region to the future of the first singularity where both solutions are defined.

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arXiv2023

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in $1D$ and multi-dimensions

In this article, we provide notes that complement the lectures on the relativistic Euler equations and shocks that were given by the second author at the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schrodinger International Institute for Mathematics and Physics in Vienna in February, 2022. We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and providing a brief overview of local well-posedness in Sobolev spaces. Then, using Riemann invariants, we provide the first detailed construction of a localized subset of the maximal globally hyperbolic developments of an open set of initially smooth, shock-forming isentropic solutions in 1D, with a focus on describing the singular boundary and the Cauchy horizon that emerges from the singularity. Next, we provide an overview of the new second-order formulation of the 3D relativistic Euler equations derived in [41], its rich geometric and analytic structures, their implications for the mathematical theory of shock waves, and their connection to the setup we use in our 1D analysis of shocks. We then highlight some key prior results on the study of shock formation and related problems. Furthermore, we provide an overview of how the formulation of the flow derived in [41] can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks.

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arXiv2023

Stable Big Bang formation for Einstein's equations: The complete sub-critical regime

For $(t,x) \in (0,\infty)\times\mathbb{T}^D$, the generalized Kasner solutions are a family of explicit solutions to various Einstein-matter systems that start out smooth but then develop a Big Bang singularity as $t \downarrow 0$, i.e., curvature blowup along a spacelike hypersurface. The family is parameterized by the Kasner exponents $\widetilde{q}_1,\cdots,\widetilde{q}_D \in \mathbb{R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, given say at $\lbrace t = 1 \rbrace$, as long as the exponents are "sub-critical" in the following sense: $\mathop{\max_{I,J,B=1,\cdots,D}}_{I < J} \{\widetilde{q}_I+\widetilde{q}_J-\widetilde{q}_B\}<1$. Previous works have shown the stability of the singularity under stronger assumptions: 1) the Einstein-scalar field system with $D = 3$ and $\widetilde{q}_1 \approx \widetilde{q}_2 \approx \widetilde{q}_3 \approx 1/3$ or 2) the Einstein-vacuum equations for $D \geq 39$ with $\max_{I=1,\cdots,D} |\widetilde{q}_I| < 1/6$. We prove that the Kasner singularity is dynamically stable for \emph{all} sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the $1+D$-dimensional Einstein-scalar field system for $D \geq3$ and the $1+D$ dimensional Einstein-vacuum equations for $D \geq 10$. Moreover, for the Einstein-vacuum equations in $1+3$ dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have stable Big Bangs under polarized $U(1)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized $U(1)$-symmetric solutions.

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arXiv2022

Rough sound waves in $3D$ compressible Euler flow with vorticity

We prove a series of results tied to the regularity and geometry of solutions to the $3D$ compressible Euler equations with vorticity and entropy. Our framework exploits and reveals additional virtues of a recent new formulation of the equations, which decomposed the flow into a geometric "(sound) wave-part" coupled to a "transport-div-curl-part" (transport-part for short), with both parts exhibiting remarkable properties. Our main result is that the time of existence can be controlled in terms of the $H^{2^+}(\mathbb{R}^3)$-norm of the wave-part of the initial data and various Sobolev and Hölder norms of the transport-part of the initial data, the latter comprising the initial vorticity and entropy. The wave-part regularity assumptions are optimal in the scale of Sobolev spaces: shocks can instantly form if one only assumes a bound for the $H^2(\mathbb{R}^3)$-norm of the wave-part of the initial data. Our proof relies on the assumption that the transport-part of the initial data is more regular than the wave-part, and we show that the additional regularity is propagated by the flow, even though the transport-part of the flow is deeply coupled to the rougher wave-part. To implement our approach, we derive several results of independent interest: i) sharp estimates for the acoustic geometry, i.e., the geometry of sound cones; ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; and iii) Schauder estimates for the transport-div-curl-part. Compared to previous works on low regularity, the main new features of the paper are that the quasilinear PDE systems under study exhibit multiple speeds of propagation and that elliptic estimates for various components of the fluid are needed, both to avoid loss of regularity and to gain space-time integrability.

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arXiv2022

The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy

Consider a $1$D simple small-amplitude solution $(ρ_{(bkg)}, v^1_{(bkg)})$ to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite time. Viewing $(ρ_{(bkg)}, v^1_{(bkg)})$ as a plane-symmetric solution to the full compressible Euler equations in $3$D, we prove that the shock-formation mechanism for the solution $(ρ_{(bkg)}, v^1_{(bkg)})$ is stable against all sufficiently small and compactly supported perturbations. In particular, these perturbations are allowed to break the symmetry and have non-trivial vorticity and variable entropy. Our approach reveals the full structure of the set of blowup-points at the first singular time: within the constant-time hypersurface of first blowup, the solution's first-order Cartesian coordinate partial derivatives blow up precisely on the zero level set of a function that measures the inverse foliation density of a family of characteristic hypersurfaces. Moreover, relative to a set of geometric coordinates constructed out of an acoustic eikonal function, the fluid solution and the inverse foliation density function remain smooth up to the shock; the blowup of the solution's Cartesian coordinate partial derivatives is caused by a degeneracy between the geometric and Cartesian coordinates, signified by the vanishing of the inverse foliation density (i.e., the intersection of the characteristics).

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arXiv2022

The emergence of the singular boundary from the crease in $3D$ compressible Euler flow

We study the Cauchy problem for the $3D$ compressible Euler equations under an arbitrary equation of state with positive speed of sound, aside from that of a Chaplygin gas. For open sets of smooth initial data with non-trivial vorticity and entropy, our main results yield a constructive proof of the formation, structure, and stability of the singular boundary, which is the set of points where the solution forms a shock singularity, i.e., where some first-order Cartesian coordinate partial derivatives of the velocity and density blow up. We prove that in the solution regime under study, the singular boundary has the structure of a degenerate, acoustically null $3D$ submanifold-with-boundary. Our approach yields the full structure of a neighborhood of a connected component of the crease, which is a $2D$ acoustically spacelike submanifold equal to the past boundary of the singular boundary. In the study of shocks, the crease plays the role of the "true initial singularity" from which the singular boundary emerges, and it is a crucial ingredient for setting up the shock development problem. These are the first results revealing the totality of these structures without symmetry, irrotationality, or isentropicity assumptions. Moreover, even within the sub-class of irrotational and isentropic solutions, these are the first constructive results revealing these structures without a strict convexity assumption on the shape of the singular boundary. Our proof relies on a new method: the construction of rough foliations of spacetime, dynamically adapted to the exact shape of the singular boundary and crease, where the latter is provably two degrees less differentiable than the fluid.

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arXiv2020

Remarkable localized integral identities for $3D$ compressible Euler flow and the double-null framework

We derive new, localized geometric integral identities for solutions to the $3D$ compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The identities are coercive in the first derivatives of the specific vorticity and the second derivatives of the entropy, and the error terms exhibit remarkable regularity and null structures. Our framework allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution's higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric and ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. Our main new contribution is our analysis of the positive co-dimension, spacelike boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation, we observe several crucial cancellations, which in total show that the boundary terms have a good sign.

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arXiv2019

The relativistic Euler equations: Remarkable null structures and regularity properties

We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and elliptic equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

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arXiv2018

On the nature of Hawking's incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking's celebrated "singularity" theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that i) it does not rely on approximate monotonicity identities and ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.

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arXiv2018

Shock formation for $2D$ quasilinear wave systems featuring multiple speeds: Blowup for the fastest wave, with non-trivial interactions up to the singularity

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. 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 (shock-forming) fast wave and therefore exhibit very poor commutation properties with the slow wave operator, much worse than their commutation properties with a transport operator. 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.

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arXiv2018

A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on $(0,\infty) \times \mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace t = 0 \rbrace$. We place initial data for the linearized system along $\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsing direction $t \downarrow 0$. Our first main result is the proof of an approximate $L^2$ monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) solution. In particular, we show that as $t \downarrow 0$, various time-rescaled components of the linear solution converge to regular functions defined along $\lbrace t = 0 \rbrace$. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well-posed only in the direction $t \downarrow 0$. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction $t \downarrow 0$ under small perturbations of its data at $\lbrace t = 1 \rbrace$.

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arXiv2017

The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology $\mathbb{S}^3$

The Friedmann--Lema\^ıtre--Robertson--Walker (FLRW) solution to the Einstein-scalar field system with spatial topology $\mathbb{S}^3$ models a universe that emanates from a singular spacelike hypersurface (the Big Bang), along which various spacetime curvature invariants blow up, only to re-collapse in a symmetric fashion in the future (the Big Crunch). In this article, we give a complete description of the maximal developments of perturbations of the FLRW data at the chronological midpoint of its evolution. We show that the perturbed solutions also exhibit curvature blowup along a pair of spacelike hypersurfaces, signifying the stability of the Big Bang and the Big Crunch. Moreover, we provide a sharp description of the asymptotic behavior of the solution up to the singularities, showing in particular that various time-rescaled solution variables converge to regular tensorfields on the singular hypersurfaces that are close to the corresponding FLRW tensorfields. Our proof crucially relies on $L^2$-type approximate monotonicity identities in the spirit of the ones we used in our joint works with Rodnianski, in which we proved similar results for nearly spatially flat solutions with spatial topology $\mathbb{T}^3$. In the present article, we rely on new ingredients to handle nearly round spatial metrics on $\mathbb{S}^3$, whose curvatures are order-unity near the initial data hypersurface. In particular, our proof relies on i) the construction of a globally defined spatial vectorfield frame adapted to the symmetries of a round metric on $\mathbb{S}^3$; ii) estimates for the Lie derivatives of various geometric quantities with respect to the elements of the frame; and iii) sharp estimates for the asymptotic behavior of the FLRW solution's scale factor near the singular hypersurfaces.

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arXiv2017

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 Ψ+ (1 + Ψ)^P ΔΨ= 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $Ψ$ is initially near $0$ while $1 + Ψ$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + Ψ$ in finite time. The solution remains regular up to the degeneracy in the following sense: there is a high-order energy, featuring degenerate weights only at the top order, that remains bounded up to the time of first vanishing. When $P=1$, we show that any $C^1$ extension of $Ψ$ to the future of a point where $1 + Ψ= 0$ must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar (which is a curvature invariant) of the Lorentzian metric corresponding to the wave equation blows up at those points. In particular, our results show that curvature blowup for the metric of a quasilinear wave equation does not always coincide with singularity formation in the solution variable. Similar phenomena occur when $P=2$, but in this case, the vanishing of $1 + Ψ$ corresponds only to a breakdown in the strict hyperbolicity of the equation.

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arXiv2017

Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearities

We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or large in $L^{\infty}$, but the spatial derivatives must initially satisfy a nonlinear smallness condition compared to the time derivative. The first main idea of the proof is to construct a quasilinear integrating factor, which allows us to reformulate the wave equation as a system whose solutions remain regular, all the way up to the singularity. This is equivalent to constructing quasilinear vectorfields adapted to the nonlinear flow. The second main idea is to exploit some crucial monotonic terms in various estimates, especially the energy estimates, that feature the integrating factor. The availability of the monotonicity is tied to our assumptions on the data and on the structure of the quasilinear terms. The third main idea is to propagate the relative smallness of the spatial derivatives all the way up to the singularity so that the solution behaves, in many ways, like an ODE solution. As a corollary of our main results, we show that there are quasilinear wave equations that exhibit two distinct kinds of blowup: the formation of shocks for one non-trivial set of data, and ODE-type blowup for another non-trivial set.

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arXiv2017

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

In $n \geq 1$ 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.

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arXiv2017

A New Formulation of the $3D$ Compressible Euler Equations With Dynamic Entropy: Remarkable Null Structures and Regularity Properties

We derive a new formulation of the $3D$ compressible Euler equations with dynamic entropy exhibiting remarkable null structures and regularity properties. Our results hold for an arbitrary equation of state (which yields the pressure in terms of the density and the entropy) in non-vacuum regions where the speed of sound is positive. Our work is an extension of our prior joint work with J. Luk, in which we derived a similar new formulation in the special case of a barotropic fluid, that is, when the equation of state depends only on the density. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity (defined to be vorticity divided by density), a transport equation for the entropy, and some additional transport-divergence-curl-type equations involving special combinations of the derivatives of the solution variables. The good geometric structures in the equations allow one to use the full power of the vectorfield method in treating the "wave part" of the system. In a forthcoming application, we will use the new formulation to give a sharp, constructive proof of finite-time shock formation, tied to the intersection of acoustic "wave" characteristics, for solutions with nontrivial vorticity and entropy at the singularity. In the present article, we derive the new formulation and overview the central role that it plays in the proof of shock formation.

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