Differential and Integral Equations

The periodic Cauchy problem for the 2-component Camassa-Holm system

Ryan C. Thompson

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 $0 \leq r <s$.

Article information

Source
Differential Integral Equations, Volume 26, Number 1/2 (2013), 155-182.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1355867512

Mathematical Reviews number (MathSciNet)
MR3058703

Zentralblatt MATH identifier
1299.35271

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Thompson, Ryan C. The periodic Cauchy problem for the 2-component Camassa-Holm system. Differential Integral Equations 26 (2013), no. 1/2, 155--182. https://projecteuclid.org/euclid.die/1355867512