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

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.