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

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