Nonuniform dependence on initial data for the 2D viscous shallow water equations

Abstract

The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system. In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations, which is a hyperbolic-parabolic system. We give a new approach to studying the issue of nonuniform dependence on initial data for these equations. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces 𝐻 𝑠 × 𝐻 𝑠 for 𝑠 > 2.