Zero-filter limit issue for the Camassa-Holm equation in Besov spaces

Abstract

In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in $L^{\infty }(0,T;B_{2,r}^s(\mathbb{R}))$ to the inviscid Burgers equation as the filter parameter $\hat{I}\pm$ tends to zero with the given initial data $u_0\in B_{2,r}^s(\mathbb{R})$. Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in $B_{2,r}^s(\mathbb{R})$.