A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$

TL;DR

This paper examines the zero-filter limit of the Camassa-Holm equation in the Besov space $B^s_{2, ext{infinity}}( ext{R})$, showing that solutions do not converge strongly to the Burgers equation in this space, contrasting previous results.

Contribution

It demonstrates that in the space $B^s_{2, ext{infinity}}( ext{R})$, solutions of the Camassa-Holm equation fail to converge strongly to the Burgers equation as the filter parameter approaches zero.

Findings

Strong convergence fails in $B^s_{2, ext{infinity}}$ space.

Contrasts with previous convergence results in $B^s_{2,r}$ spaces.

Highlights the importance of the choice of function space for limit behavior.

Abstract

This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data $u_0\in B^s_{2,r}(\R)$ with $s>\frac32$ and $1\leq r<\infty$, the solutions $\mathbf{S}_{t}^{\mathbf{\alpha}}(u_0)$ of the Camassa-Holm equation exhibit convergence in the $L^\infty_T(B^s_{2,r})$ norm to the unique solution of the Burgers equation as $\alpha\rightarrow 0$. Contrary to this result, the present study demonstrates that for initial data $u_0\in B^s_{2,\infty}(\R)$ the solutions of the Camassa-Holm equation fail to converge strongly in the $L^\infty_T(B^s_{2,\infty})$ norm to the Burgers equation as $\alpha\rightarrow 0$.