Global solvability to the 3D incompressible magneto-micropolar system with vacuum. (English) Zbl 1498.35432
Summary: 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
\[
\begin{gathered}
( \| \sqrt{\rho_0 u_0} \|_{L^2}^2 + \| \sqrt{\rho_0} w_0 \|_{L^2}^2 + \|b_0 \|_{L^2}^2) \times ( \mu_1 \| \nabla u_0\|_{L^2}^2 + \mu_2 \| \nabla w_0\|_{L^2}^2 \\
+ ( \mu_2 + \lambda) \| \mathrm{div} w_0\|_{L^2}^2 + \eta \| \nabla b_0 \|_{L^2}^2 + \xi \| 2 w_0 - \nabla \times u_0 \|_{L^2}^2)
\end{gathered}
\]
is suitably small, which extends the corresponding result of F. W. Cruz and M. M. Novais [Appl. Anal. 101, No. 6, 1963–1970 (2022; Zbl 1487.35284)] to the inhomogeneous case, and Z. Ye’s result [Discrete Contin. Dyn. Syst., Ser. B 24, No. 12, 6725–6743 (2019; Zbl 1428.35407)] to the 3D Cauchy problem of the inhomogeneous micropolar equations with magnetic field. Furthermore, we also established the large time behavior of strong solutions.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76A05 | Non-Newtonian fluids |
| 76U05 | General theory of rotating fluids |
| 35D35 | Strong solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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