Research On Regularity To The Weak Solutions Of The Incompressible Magneto-micropolar Equations

In this paper, the regularity of the weak solutions to the3D incompressible magneto-micropolar equations is studied.where u=（u₁（x, t）, u₂（x, t）, u3（x, t）） denotes the velcity of the fluid at a point （x, t）∈R³×[0, T), ω=（ω₁（x, t）, ω₂（x, t）, ω₃（x, t））, b=（b₁（x, t）, b₂（x, t）, b₃（x, t）） and p=p（x, t） denote,respectively, the micro-rotational velocity, the magnetic field and the hydrostatic pressure.u0（x）, ω0（x） and b0（x） are the prescribed initial data for the velocity, angular velocity andmagnetic field with properties divu0=0and divb0=0. μ is the kinematic viscosity, χ is thevortex viscosity, κ and γ are spin viscosities,1νis the magnetic Reynold number.If b=0, Equations（0.0.1） reduce to the incompressible micropolar fluid system:Clearly, if both ω=0and χ=0, then Equations（0.0.1） reduce to be the incompressiblemagneto-hydrodynamic equations:If, further, ω=0, b=0and χ=0,（0.0.1） reduces to the classical Navier-Stokes equations:In this paper, we study the continuity criteria of smooth solutions to the incompressiblemagneto-micropolar fluid equations in R3by energy methods. Combining Giga’s[13]clas-sical local existence theorem of strong solution for semilinear parabolic equations and theSerrin’s[27]uniqueness criterion, we conclude that the weak solution is regular. Our results asfollows:1) If u_z∈L^q（0, T; L^p（R³））, with3/p+2/q≤1, p≥3,then the weak solution （u, ω, b）is regular on （0, T）.2) If pressure P（=p+b²）∈L^q（0, T; L^p（R³））, with3/p+3/q≤2,3/2<p≤∞, thenthe weak solution （u, ω, b） is regular on （0, T）.3) If P_z∈L^q（0, T; Lp（R³）） with3/p+2/q≤7/4,12/7≤p,and12/7≤p≤4, then the weaksolution （u, ω, b） is regular on （0, T）.