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

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    * Corresponding author
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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  • 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.

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


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