Smooth and stable Euler implosions

TL;DR

This paper introduces a new class of smooth, non-isentropic implosion profiles for multi-dimensional Euler equations, proving their stability under certain perturbations and providing explicit similarity exponents.

Contribution

It constructs explicit self-similar implosion profiles for the Euler equations and analyzes their nonlinear stability under various perturbations.

Findings

Constructed explicit self-similar implosion profiles.

Proved stability of the ground state solution to radially symmetric perturbations.

Characterized initial data sets leading to nonlinear stability for non-symmetric perturbations.

Abstract

We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.