Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

Lei Zhang; Bin Liu

doi:10.3934/dcds.2018112

Article Contents

2018, Volume 38, Issue 5: 2655-2685. Doi: 10.3934/dcds.2018112

This issue Previous Article Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models Next Article

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

Lei Zhang^, and
Bin Liu

School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

* Corresponding author
* Corresponding author

Received: June 30, 2017

Revised: November 30, 2017

Published: June 2018

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11571126 and No. 11701198, the China Postdoctoral Science Foundation funded project under Grant No. 2017M622397

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Abstract

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in $ B_{p,r}^s× B_{p,r}^{s-1}$ with $s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$, $p,r∈ [1,∞]$ by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space $ B_{2,1}^{3/2}× B_{2,1}^{1/2}$, and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

Keywords:

Rotation-two-component Camassa-Holm system,
critical Besov space,
blow-up phenomena,
Gevery regularity and analyticity.

Mathematics Subject Classification: Primary: 35Q35, 35L35, 35G55, 35A10.

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