## Non-uniform dependence on initial data for a family of non-linear evolution equations.(English)Zbl 1212.35426

Summary: We show that solutions to the periodic Cauchy problem for a family of non-linear evolution equations, which contains the Camassa-Holm equation, do not depend uniformly continuously on initial data in the Sobolev space $$H^s(\mathbb T)$$, when $$s=1$$ or $$s\geq 2$$.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35B65 Smoothness and regularity of solutions to PDEs

### Keywords:

periodic Cauchy problem; Camassa-Holm equation