Nonuniform dependence on initial data for compressible gas dynamics: The periodic Cauchy problem

TL;DR

This paper investigates the dependence of solutions to the periodic Cauchy problem for ideal compressible gas dynamics on initial data, revealing nonuniform dependence despite established well-posedness.

Contribution

It demonstrates that the data-to-solution map is continuous but not uniformly continuous in Sobolev spaces for the two-dimensional periodic case.

Findings

The Cauchy problem is well-posed with continuous dependence on initial data.

The data-to-solution map lacks uniform continuity in Sobolev spaces.

This nonuniform dependence highlights subtle stability issues in gas dynamics models.

Abstract

We start with the classic result that the Cauchy problem for ideal compressible gas dynamics is locally well posed in time in the sense of Hadamard; there is a unique solution that depends continuously on initial data in Sobolev space $H^s$ for $s>d/2+1$ where $d$ is the space dimension. We prove that the data to solution map for periodic data in two dimensions although continuous is not uniformly continuous on any bounded subset of Sobolev class functions.