Non-uniform dependence on initial data for the 2D viscous shallow water equations

TL;DR

This paper demonstrates that the solution map for the 2D viscous shallow water equations is not uniformly continuous in Sobolev spaces, highlighting a nuanced dependence on initial data in hyperbolic-parabolic systems.

Contribution

It proves the non-uniform dependence on initial data for the 2D viscous shallow water equations in Sobolev spaces, extending understanding of solution stability.

Findings

Solution map is not uniformly continuous in Sobolev spaces for s>2

Highlights complex dependence of solutions on initial data

Provides insight into stability issues of hyperbolic-parabolic systems

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 prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s\times H^{s}$ for $s>2$.