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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ (\rho_0, u_0, w_0, b_0) $\end{document}</tex-math></inline-formula> satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is suitably small, which extends the corresponding Cruz and Novais's result (Appl. Anal., 2020[<xref ref-type="bibr" rid="b9">9</xref>]) to the inhomogeneous case, and Ye's result (Discrete Contin. Dyn. Syst. B, 2019[<xref ref-type="bibr" rid="b17">17</xref>]) to the 3D Cauchy problem of the inhomogeneous micropolar equations with magnetic field. Furthermore, we also established the large time behavior of strong solutions.</p>