Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

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 transport-div-curl 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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“…Covariant wave equation for the logarithmic enthalpy. We now derive the covariant wave equation (37).…”

Section: 2mentioning

confidence: 99%

The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

Disconzi

Speck

2019

Ann. Henri Poincaré

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“…We plan to study a convenient open set of initial conditions in three spatial dimensions whose solutions typically have non-zero vorticity and non-constant entropy: perturbations (without symmetry conditions) of simple isentropic (that is, constant entropy 24 ) plane waves. 25 We note that in our joint work [27] on scalar wave equations in two spatial dimensions, we proved shock formation for solutions corresponding to a similar set of nearly plane symmetric initial data. The advantage of studying perturbations of simple isentropic plane waves is that it allows us to focus our attention on the singularity formation without having to confront additional evolutionary phenomena that are often found in solutions to wave-like systems.…”

Section: Overview Of the Roles Of Theorems 31 And 32 In Proving Shomentioning

confidence: 74%

“…The first use of an eikonal function in proving a global result for a nonlinear hyperbolic system occurred in the celebrated proof [6] of the stability of the Minkowski spacetime as a solution to the Einstein-vacuum equations. 27 Eikonal functions have also played a central role in proofs of low-regularity well-posedness for quasilinear hyperbolic equations, most notably the recent Klainerman-Rodnianski-Szeftel proof of the bounded L 2 curvature conjecture [18].…”

Section: 2mentioning

confidence: 99%

“…Step (6)). These aspects of the proof, though of fundamental importance, have been well-understood since Christodoulou's work [5] and are described in more detail in [19,20,27]; for this reason, we will not further discuss these issues here. 36 With the help of the vectorfield multiplier method, based on the energy-momentum tensor for wave equations and the multiplier vectorfield T := (1 + 2µ)L + 2X; see [19,20,27].…”

Section: 3mentioning

confidence: 99%

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

Speck

2019

Arch Rational Mech Anal

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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.Although the equations are significantly more complicated than they are in the barotropic case, they enjoy many of same remarkable features including i) all derivative-quadratic inhomogeneous terms are null forms relative to the acoustical metric, which is the Lorentzian metric driving the propagation of sound waves and ii) the transport-divergence-curl-type equations allow one to show that the entropy is one degree more differentiable than the velocity and that the vorticity is exactly as differentiable as the velocity, which represents a gain of one derivative compared to standard estimates. This gain of a derivative, which seems to be new for the entropy, is essential for closing the energy estimates in our forthcoming proof of shock formation since the second derivatives of the entropy and the first derivatives of the vorticity appear as inhomogeneous terms in the wave equations. 1For sufficiently regular solutions, there are many equivalently formulations of the compressible Euler equations, depending on the state-space variables that one chooses as unknowns in the system.2 In our forthcoming proof of shock formation, we will, for convenience, consider spacetimes with topology R × Σ, where Σ := R × T 2 is the space manifold; see Sect. 4 for an overview. In that context, {x α } α=0,1,2,3 denotes the usual Cartesian coordinate system on R × Σ, where x 0 ∈ R is the time coordinate, x 1 is a standard spatial coordinate on R, and x 2 and x 3 are standard (locally defined) coordinates on T 2 . Note that the vectorfields ∂ α := ∂ ∂x α on T 2 can...

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“…In the wake of the above results, there have been significant further advancements, which we now describe. In [38], we extended the shock formation results of [37] 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 [38], the solutions do not experience dispersive decay.…”

Section: 72mentioning

confidence: 97%

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

Speck

2019

Pure Appl. Analysis

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In n ≥ 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 firstorder 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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Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Cited by 30 publications

References 64 publications

The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

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

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

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