Nonuniform dependence on initial data for compressible gas dynamics: The Cauchy problem on $\mathbb{R}^2$

TL;DR

This paper demonstrates that for the 2D compressible Euler equations, the data-to-solution map is continuous but not uniformly continuous in Sobolev spaces, revealing nuanced dependence on initial data.

Contribution

It establishes the non-uniform dependence of solutions on initial data for the 2D compressible Euler equations in Sobolev spaces, highlighting a subtle aspect of well-posedness.

Findings

Data-to-solution map is continuous in Sobolev spaces.

The map is not uniformly continuous on bounded subsets.

Results apply to the Cauchy problem on .

Abstract

The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is continuous. We prove that the data-to-solution map on the plane is not uniformly continuous on any bounded subset of Sobolev class functions.