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 Hs 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.