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

Jonathan Luk and Jared Speck

Vol. 17 (2024), No. 3, 831–941
Abstract

Consider a one-dimensional simple small-amplitude solution (ϱ(bkg ),v(bkg )1) 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(bkg )1) as a plane-symmetric solution to the full compressible Euler equations in three dimensions, we prove that the shock-formation mechanism for the solution (ϱ(bkg ),v(bkg )1) is stable against all sufficiently small and compactly supported perturbations. In particular, these perturbations are allowed to break the symmetry and have nontrivial 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).

Keywords
compressible Euler equations, shock formation, stable singularity formation, wave breaking, vectorfield method, characteristics, eikonal function, null condition, null hypersurface, null structure
Mathematical Subject Classification
Primary: 35L67
Secondary: 35L05, 35Q31, 76N10
Milestones
Received: 5 September 2021
Revised: 30 June 2022
Accepted: 3 August 2022
Published: 24 April 2024
Authors
Jonathan Luk
Stanford University
Stanford, CA
United States
Jared Speck
Vanderbilt University
Nashville, TN
United States

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