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

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.