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 \(L^\infty (0,T;B^s_{2,r}(\mathbb {R}))\) to the inviscid Burgers equation as the filter parameter \(\alpha \) tends to zero with the given initial data \(u_0\in B^s_{2,r}(\mathbb {R})\). 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 \(B^s_{2,r}(\mathbb {R})\).