Abstract
This paper focus on the Cauchy problem of the 3D incompressible magneto-micropolar equations with fractional dissipation in the Sobolev space. Liu, Sun and Xin obtained the global solutions to the 3D magneto-micropolar equations with $\alpha=\beta=\gamma=\frac{5}{4}$. Deng and Shang established the global well-posedness of the 3D magneto-micropolar equations in the case of $\alpha\geq\frac{5}{4}$, $\alpha+\beta\geq\frac{5}{2}$ and $\gamma\geq2-\alpha\geq\frac{3}{4}$. In this paper, we establish the global well-posedness of the 3D magneto-micropolar equations with $\alpha=\beta=\frac{5}{4}$ and $\gamma=\frac{1}{2}$, which improves the results of Liu-Sun-Xin and Deng-Shang by reducing the value of $\gamma$ to $\frac{1}{2}$.
Highlights
We are concerned with the global well-posedness to the 3D magneto-micropolar equations with fractional dissipation
Many more exciting results on the global regularity for magneto-micropolar equations with partial dissipation are available for the 2D case
By the Gronwall’s inequality, (3.3) and (3.18), we get the uniqueness immediately
Summary
We are concerned with the global well-posedness to the 3D magneto-micropolar equations with fractional dissipation. Many more exciting results on the global regularity for magneto-micropolar equations with partial dissipation are available for the 2D case (see for example [9, 14, 16, 31, 32]). For the 3D case of the system (1.1), Yuan [28] first established the regularity of weak solutions and blow-up criteria for smooth solutions in the whole space. Li-Shang [8] and Tan-Wu [20] established the existence of 3D small global smooth solutions in the case of α = β = γ = 1. Deng and Shang [2] established the global well-posedness of magneto-micropolar equations n 4.
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