Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity

Xuemin Deng; Yuelong Xiao; Aibin Zang

doi:10.3934/cpaa.2021018

Article Contents

2021, Volume 20, Issue 3: 1229-1240. Doi: 10.3934/cpaa.2021018

This issue Previous Article Existence of solution and asymptotic behavior for a class of parabolic equations Next Article Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders

Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity

1.
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411100, China
2.
The Center of Applied Mathematics, Yichun University, Yichun, Jiangxi, 336000, China

^* Corresponding author
^* Corresponding author

Received: August 2020

Revised: December 2020

Early access: February 2021

Published: March 2021

The work of Aibin Zang is partially supported by School of Mathematics and Computational Science in Xiangtan University as he visited the second author and supported in part by National Natural Science Foundation of China (Grant no. 11771382). The research of Yuelong Xiao is partially supported by National Natural Science Foundation of China (Grant no. 11871412, 11771300). The study of Xuemin Deng Supported by Hunan Provincial Innovation Foundation For Postgraduate (Grant no. XDCX2021B096)

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Abstract

In this paper, we start to investigate the global existence and uniqueness of weak solutions of the $ n $-dimensional ($ n\geq3 $) hyper-dissipative Boussinesq system without thermal diffusivity in the periodic domain $ \mathbb{T}^n $ with the initial data $ u_0\in L^2(\mathbb{T}^n) $ and $ \theta_0 \in L^2(\mathbb{T}^n)\cap L^{\frac{4n}{n+2}}(\mathbb{T}^n) $. Then we focus on the vanishing thermal diffusion limit and obtain the convergent result in the sense of $ L^2 $-norm. Ultimately, we also prove the global regularity of this system in the case of 3-dimension.

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Mathematics Subject Classification: Primary: 35P05, 35Q35.

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References

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