Non-uniform convergence of solution for the Camassa-Holm equation in the zero-filter limit

TL;DR

This paper demonstrates that solutions of the Camassa-Holm equation do not converge uniformly to the inviscid Burgers equation in the zero-filter limit for certain initial data, highlighting non-uniform convergence behavior.

Contribution

It provides a proof of non-uniform convergence of Camassa-Holm solutions to Burgers solutions as the filter parameter approaches zero, extending previous results.

Findings

Solutions do not converge uniformly in the specified function space.

Non-uniform convergence persists for initial data in H^s with s > 3/2.

Results supplement earlier findings on the zero-filter limit.

Abstract

In the short note, we prove that given initial data $\mathcal{u}_0 \in \pmb{H}^s(\mathbb{R})$ with $s>\frac32$ and for some $T>0$, the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in $\pmb{L}^\infty$ $(0,T;H^s(\mathbb{R}))$ to the inviscid Burgers equation as the filter parameter $\alpha$ tends to zero. This is a supplement to our recent result on the zero-filter limit.