Expanding large global solutions of the equations of compressible fluid mechanics

TL;DR

This paper proves the nonlinear stability of expanding affine solutions to the 3D compressible Euler equations with physical vacuum, showing small perturbations remain smooth and do not form shocks for certain adiabatic exponents.

Contribution

It introduces a new interpretation of affine motions via an (almost) invariant GL$^+(3)$ action and develops a novel high-order energy framework to establish global stability.

Findings

Affine solutions are nonlinearly stable for $oldsymbol{\gamma ext{ in }(1,rac{5}{3}]}$.

Small perturbations lead to globally smooth solutions that stay close to affine motions.

No shock formation occurs in the perturbed solutions.

Abstract

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the three-dimensional isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent $\gamma$ belongs to the interval $(1,\frac53]$, then these affine motions are nonlinearly stable. Small perturbations lead to globally-in-time defined solutions that remain in the vicinity of the manifold of affine motions, they remain smooth in the interior of their support, and no shocks are formed in the process. Our strategy relies on two key ingredients. We first provide a new interpretation of the affine motions using an (almost) invariant action of GL$^+(3)$ on the compressible Euler system. This transformation dictates a particular rescaling of time and a change of variables, which in turn exposes a stabilization mechanism induced by the expansion of the background affine motions when $\gamma\in(1,\frac53]$. We then switch to a Lagrangian description of the rescaled Euler system which reflects the geometry of expanding affine motions. We introduce new ideas with respect to the existing well-posedness frameworks and build high-order energy spaces to prove global-in-time stability, thereby making crucial use of the new stabilization effect