American Institute of Mathematical Sciences

May  2018, 38(5): 2655-2685. doi: 10.3934/dcds.2018112

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

 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

Received  July 2017 Revised  December 2017 Published  March 2018

Fund Project: 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.

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.

Citation: Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
References:

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