Formation of unstable shocks for 2D isentropic compressible Euler

TL;DR

This paper constructs and analyzes unstable shock solutions in 2D isentropic compressible Euler equations with azimuthal symmetry, demonstrating convergence to a self-similar solution and stability properties.

Contribution

It introduces a novel construction of unstable shocks in 2D Euler equations using modulation and Newton methods, with detailed stability analysis.

Findings

Initial data converges to the $C^{1/5}$ self-similar solution.

Behavior is stable in $C^8$ modulo a 2D linear subspace.

Method combines modulation variables with a Newton scheme.

Abstract

In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler in azimuthal symmetry. More specifically, we construct initial data that when viewed in self-similar coordinates, converges asymptotically to the unstable $C^{\frac15}$ self-similar solution to the Burgers' equation. Moreover, we show the behavior is stable in $C^8$ modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding manifold of initial data leading to stability: rather, we rely on modulation variable techniques in conjunction with a Newton scheme.