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

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 $u_0\in B_{2,r}^s(\mathbb{R})$ with $s>\frac{3}{2}$ and $1\le r<\infty$, the solutions ${\mathbf{S}}_t^{\alpha }(u_0)$ of the Camassa-Holm equation exhibit convergence in the $L_T^{\infty }(B_{2,r}^s)$ norm to the unique solution of the Burgers equation as $\alpha \to 0$. Contrary to this result, the present study demonstrates that for initial data $u_0\in B_{2,\infty }^s(\mathbb{R})$ the solutions of the Camassa-Holm equation fail to converge strongly in the $L_T^{\infty }(B_{2,\infty }^s)$ norm to the Burgers equation as $\alpha \to 0$.

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