The periodic Cauchy problem for the 2-component Camassa-Holm system. (English) Zbl 1299.35271

Thompson, Ryan C.

Differ. Integral Equ. 26, No. 1-2, 155-182 (2013).

Summary: For Sobolev exponent \(s>3/2\), it is shown that the data-to-solution map for the 2-component Camassa-Holm system is continuous from \(H^s\times H^{s-1}(\mathbb {T})\) into \(C([0,T];H^s\times H^{s-1}(\mathbb {T}))\) but not uniformly continuous. The proof of non-uniform dependence on the initial data is based on the method of approximate solutions, delicate commutator and multiplier estimates, and well-posedness results for the solution and its lifespan. Also, the solution map is Hölder continuous if the \(H^s\times H^{s-1}(\mathbb {T})\) norm is replaced by an \(H^r\times H^{r-1}(\mathbb {T})\) norm for \(0\leq r<s\).

MSC:

35Q53

KdV equations (Korteweg-de Vries equations)

Keywords:

Camassa-Holm system; Cauchy problem; Hölder continuous; well-posedness