Continuity for the rotation-two-component Camassa-Holm system

Chenghua Wang; Rong Zeng; Shouming Zhou; Bin Wang; Chunlai Mu

doi:10.3934/dcdsb.2019160

Article Contents

2019, Volume 24, Issue 12: 6633-6652. Doi: 10.3934/dcdsb.2019160

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Continuity for the rotation-two-component Camassa-Holm system

1.
School of Public Affairs, Chongqing University, Chongqing 400044, China
2.
Department of Mathematics, Southwestern University of Finance and Economics, Sichuan 611130, China
3.
College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
4.
College of Mathematics and statistics, Chongqing University, Chongqing 401331, China

^* Corresponding author: Shouming Zhou and Rong Zeng
^* Corresponding author: Shouming Zhou and Rong Zeng

Received: April 30, 2018

Revised: October 31, 2018

Early access: July 2019

Published: September 2019

This work is supported by the National Science Fund of China (Grant No. 11771063), Science and Technology Research Program of Chongqing Municipal Educational Committee (Grant No. KJ16003162016 and KJZD-M 201800501), Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0116). The work of Mu is supported in part by NSFC under grants 11771062, the Basic and Advanced Research Project of CQC-STC under grant cstc2015jcyjBX0007 and the Fundamental Research Funds for the Central Universities under grant 106112016CDJXZ238826

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Abstract

This paper focuses on the Cauchy problem of the rotation-two-component Camassa-Holm(R2CH) system, which is a model of equatorial water waves that includes the effect of the Coriolis force. It has been shown that the R2CH system is well-posed in Sobolev spaces $ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ with $ s>3/2 $. Using the method of approximate solutions in conjunction with well-posedness estimates, we further proved that the dependence on initial data is sharp, i.e., the data-to-solution map is continuous but not uniformly continuous. Moreover, we obtain that the solution map for the R2CH system is Hölder continuous in $ H^\theta(\mathbb{R})\times H^{\theta-1}(\mathbb{R}) $-topology for all $ 0\leq\theta<s $ with exponent $ \gamma $ depending on $ s $ and $ \theta $. The Coriolis term and higher nonlinear term in the R2CH system bring challenges to construct the counter-approximate solutions.

Keywords:

Rotation-two-component Camassa-Holm system,
Sobolev space,
non-uniform dependence,
Hölder continuity.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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References

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