Nonuniform continuity of the solution map to the two component Camassa-Holm system

We prove that the solution map of the two-component Camassa-Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space Hs(T)×Hr(T) to C([0,1],Hs(T)×Hr(T)) when s⩾1 and r⩾0. We also demonstrate the nonuniform continuous property in the continuous function space C1(T)×C1(T).