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

Abstract

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 Hölder continuous in . It seems that the non-uniform dependence on initial data in periodic Besov spaces has not appeared in the previous literature.

Keywords:

• Camassa–Holm equation
• periodic boundary value problem
• non-uniform dependence
• Besov spaces

AMS Subject Classifications:

• 35B30
• 35Q53

Acknowledgements

The authors would like to express their great gratitude to the referees for their valuable suggestions, which have led to a meaningful improvement of the paper. This work is supported by the National Natural Science Foundation of China (No. 11171115).