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 ∈ H s ( R ) with s > 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 ∞ ( 0 , T ; H s ( R ) ) to the inviscid Burgers equation as the filter parameter α tends to zero. This is a complement of our recent result on the zero-filter limit.