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

Jonathan Luk, Jared Speck

Abstract

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).

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

Abstract

Consider a D simple small-amplitude solution 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 as a plane-symmetric solution to the full compressible Euler equations in D, we prove that the shock-formation mechanism for the solution 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).

Paper Structure

This paper contains 92 sections, 72 theorems, 333 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ideas of the proof
  3. The Christodoulou theory
  4. The nearly simple plane-symmetric regime
  5. The reformulation of the equations
  6. The remarkable null structure of the reformulation
  7. Estimates for the transported variables
  8. Elliptic estimates for the vorticity and the entropy gradient
  9. $L^2$ estimates for the transport variables and the high order blowup-rate
  10. Related works
  11. Shock formation in one spatial dimension
  12. Multi-dimensional shock formation for quasilinear wave equations
  13. Multi-dimensional shock formation for the compressible Euler equations
  14. Shock development problem
  15. Other singularities for the compressible Euler equations
  16. ...and 77 more sections

Key Result

Theorem \oldthetheorem

Consider a plane-symmetric, shock-forming background solution $(\varrho_{(bkg)}, v^i_{(bkg)},s_{(bkg)})$ satisfying (1)--(6) above, where the parameter $\mathring{\upalpha}$ from point (4) is small. Consider a small perturbation of the initial data of this background solution satisfying the followin Then the corresponding unique perturbed solution satisfies the following:

Figures (2)

  • Figure 1: The dynamic vectorfield frame at two distinct points on $\mathcal{F}_u$ with the $Z$-direction suppressed, and the integral curves of the transport operator $B$ for the specific vorticity and entropy
  • Figure 2: The spacetime region and various subsets.

Theorems & Definitions (193)

  • Theorem \oldthetheorem: Main theorem -- Rough version
  • Remark \oldthetheorem
  • Remark \oldthetheorem: Results building up towards Theorem \ref{['thm:intro.main']}
  • Remark \oldthetheorem: Blowup and boundedness of quantities involving higher derivatives
  • Remark \oldthetheorem: Additional information on sub-classes of solutions
  • Remark \oldthetheorem: The maximal smooth development
  • Remark \oldthetheorem: No universal blowup-profile
  • Remark \oldthetheorem: The relativistic case
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 183 more