Formation of point shocks for 3D compressible Euler

TL;DR

This paper constructs a rigorous proof of shock formation in 3D compressible Euler equations from smooth initial data, detailing the blow-up behavior and stability of the resulting shock without symmetry assumptions.

Contribution

It provides a constructive, explicit demonstration of shock formation with vorticity and no symmetry, including precise blow-up time, location, and profile stability.

Findings

Shock forms in finite time from smooth initial data.

Blow-up profile is stable and explicitly characterized.

Solutions are smooth except at a cusp point with Hölder regularity.

Abstract

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev-class initial data which are a small $L^ \infty $ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blow up time and location can be explicitly computed, and solutions at the blow up time are smooth except for a single point, where they are of cusp-type with H\"{o}lder $C^ {\frac{1}{3}}$ regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blow up profile, necessary to establish the stability in self-similar variables of the generic shock profile.