Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude

TL;DR

This paper demonstrates that the flow map for a shallow water wave model is not uniformly continuous in Sobolev spaces, highlighting limitations in the stability of solutions with respect to initial data.

Contribution

It proves non-uniform continuity of the flow map for a shallow water wave model in Sobolev spaces, using explicit solution approximations and Sobolev norm estimates.

Findings

Flow map is not uniformly continuous in $H^s$ for $s>3/2$.

Sequences of initial data can be close in $H^s$ but lead to solutions separated at positive times.

Explicit formulae are used to approximate solutions and analyze their Sobolev norms.

Abstract

We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space $H^s$ with $s>3/2$. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in $H^s$, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in $H^s$ on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in $H^s$ at any positive time in this interval. The result is obtained by approximating the solutions corresponding to these initial data by explicit formulae and by estimating the approximation error in suitable Sobolev norms.