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