The periodic Cauchy problem for the 2-component Camassa-Holm system
2013; Khayyam Publishing; Volume: 26; Issue: 1/2 Linguagem: Inglês
10.57262/die/1355867512
ISSN0893-4983
Autores Tópico(s)Algebraic structures and combinatorial models
ResumoFor 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 $0 \leq r <s$.
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