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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
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