January/February 2013 The periodic Cauchy problem for the 2-component Camassa-Holm system
Ryan C. Thompson
Differential Integral Equations 26(1/2): 155-182 (January/February 2013). DOI: 10.57262/die/1355867512

Abstract

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 $0 \leq r <s$.

Citation

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Ryan C. Thompson. "The periodic Cauchy problem for the 2-component Camassa-Holm system." Differential Integral Equations 26 (1/2) 155 - 182, January/February 2013. https://doi.org/10.57262/die/1355867512

Information

Published: January/February 2013
First available in Project Euclid: 18 December 2012

zbMATH: 1299.35271
MathSciNet: MR3058703
Digital Object Identifier: 10.57262/die/1355867512

Subjects:
Primary: 35Q53

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 1/2 • January/February 2013
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