Global Regularity Criterion for the Magneto-Micropolar Fluid Equations
نویسنده
چکیده
ω(t, x) ∈ Rand b(t, x) ∈ R, and p(t, x) ∈ R denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. μ, χ, κ, γ, and ] are positive numbers associated with properties of the material: μ is the kinematic viscosity, χ is the vortex viscosity, κ and γ are spin viscosities, and 1/] is the magnetic Reynold. u 0 , ω 0 , and b 0 are initial data for the velocity, the angular velocity, and the magnetic field with properties div u 0 = 0 and div b 0 = 0. It is well known that the question of global existence or finite time blowup of smooth solutions for the 3D incompressible Euler orNavier-Stokes equations has been one of the most outstanding open problems in applied analysis, as well as that for the 3D incompressible magneto-micropolar fluid equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the global regularity criterion of the solutions for system (1). But there are few theories about regularity and blow-up criteria of magneto-micropolar fluid equations. Some blow-up criterion are obtained by Yuan [1] in 2010. His paper implies that most classical blow-up criteria of smooth solutions toNavierStokes or magneto-hydrodynamic equations also hold for magneto-micropolar fluid equations. In particular, using Fourier frequency localization, Yuan proved the Beale-KatoMajda criterion only relying on ∇u; that is, if
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Zhihao Tang, GangWang, and Haiwa Guan 1 Department of Fundamentals, Henan Polytechnic Institute, Nanyang, Henan 473009, China 2 Shandong Transport Vocational College, Weifang, Shandong 261206, China 3Department of Public Teaching, Wenzhou Vocational College of Science and Technology, Wenzhou, Zhejiang 325000, China Correspondence should be addressed to Haiwa Guan; [email protected] Received...
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