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Global solvability to the 3D incompressible magneto-micropolar system with vacuum

  • *Corresponding author: Yang Liu

    *Corresponding author: Yang Liu 

The first author is supported by NSF of China (11901288), Postdoctoral Science Foundation of China (2021M691219), Scientific Research Foundation of Jilin Provincial Education Department (JJKH20210 873KJ), and Natural Science Foundation of Changchun Normal University

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  • This paper deals with the Cauchy problem of 3D innhomogeneous incompressible magneto-micropolar system. We prove the global existence of strong solutions to this system, with initial data being of small norm but allowed to have vacuum and large oscillations. More precisely, we only require that the initial data $ (\rho_0, u_0, w_0, b_0) $ satisfying

    $ \begin{align*} &\Big(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|\sqrt{\rho_0}w_0\|_{L^2}^2+\|b_0\|_{L^2}^2\Big)\times\Big(\mu_1\|\nabla u_0\|_{L^2}^2 +\mu_2\|\nabla w_0\|_{L^2}^2\nonumber\\ &\quad+(\mu_2+\lambda)\|{\rm div}w_0\|_{L^2}^2+\eta\|\nabla b_0\|_{L^2}^2 +\xi\|2w_0-\nabla\times u_0\|_{L^2}^2\Big) \end{align*} $

    is suitably small, which extends the corresponding Cruz and Novais's result (Appl. Anal., 2020[9]) to the inhomogeneous case, and Ye's result (Discrete Contin. Dyn. Syst. B, 2019[17]) to the 3D Cauchy problem of the inhomogeneous micropolar equations with magnetic field. Furthermore, we also established the large time behavior of strong solutions.

    Mathematics Subject Classification: 35Q35, 76D03.


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