Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit

In this short note, we prove that given initial data \(u_0\in H^s(\mathbb {R})\) $u_0\in H^s(\mathbb{R})$ with \(s>\frac{3}{2}\) $s>\frac{3}{2}$ and for some \(T>0\) $T>0$ , the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in \(L^\infty (0,T;H^s(\mathbb {R}))\) $L^{\infty }(0,T;H^s(\mathbb{R}))$ to the inviscid Burgers equation as the filter parameter \(\alpha \) $\alpha$ tends to zero. This is a complement of our recent result on the zero-filter limit.