A note on the solution map for the periodic Camassa–Holm equation

In this paper, we study the dependence on initial data of solutions to Camassa–Holm equation with periodic boundary condition in Besov spaces. We show that when and , the solution map is not uniformly continuous from into for or from into for . Moreover, we prove that if a weaker -topology is used, then the solution map becomes Holder continuous in . It seems that the non-uniform dependence on initial data in periodic Besov spaces has not appeared in the previous literature.