Global Well-posedness Of The 3D Incompressible Magneto-micropolar Equations

The magneto micropolar fluid equation describes the motion law of micropolar fluid under the action of magnetic field,and this equation is widely used in the fields of hydrodynamics and meteorology.In recent years,many scholars have focused on the related problems of the 3D incompressible magneto micropolar fluid equations,but the well-posedness of the equations and the global regularity of weak solutions need to be further solved.In this dissertation,we focus on the global well-posedness of the 3D incompressible magneto micropolar fluid equations.First of all,we introduce the equations of the 3D incompressible magneto micropolar fluid.In this dissertation,we consider the global well-posedness of the magneto-micropolar equations which can be written as where u=(u1(x,t),u2(x,t),u3(x,t))represents the velocity field of the fluid,ω=(ω1(x,t),ω2(x,t),ω3(x,t))denotes the micro-rotational velocity,π=π(x,t)denotes the scalar pressure and b=(b1(x,t),b2(x,t),b3(x,t))is the magnetic field.The positive parameters μ is the kinematic viscosity,χ is the vortex viscosity,v is the reciprocal of the magnetic Reynolds number and κ,η are the the spin viscosities.In chapter three,we study the regularity criteria for weak solution of the 3D incompressible magneto-micropolar fluid equations(0.0.3).Using the standard energy method and the fundamental inequalities,combining with the spatial embedding relations,we obtain the extension criterion of strong solution and the regularity criterion of weak solution.Specifically,if the weak solution(u,ω,b)satisfies that for some i,j ∈ {1,2,3}with i≠j,?iui,?juj,?ibi,?jbj∈Lp(0,T;Bq,∞0(R3)),2/p+3/q=2,3≤q≤∞,then the weak solution(u,ω,b)of the equations(0.0.3)is regular on the interval(0,T].In addition,if we ignore the influence of the magnetic field b in the equations(0.0.3),then the equations(0.0.3)reduces to the micropolar equations,it is easy to verify that the above regularity criterion also holds for the micropolar fluid equations.In chapter four,we consider the following 3D generalized incompressible magnetomicropolar equations with fractional Laplacian dissipation:Here α,β,γ are positive parameters,the fractional Laplacian operator Λ2α=(-Δ)α is defined through a Fourier transform,namely We establish the global well-posedness of the system(0.0.4).By employing commutator estimates and energy method,combining with fundamental inequalities,we get that when α=β=5/4 and γ=1/2,there exists a unique global strong solution(u,ω,b)satisfying,for any T>0,(u,ω,b)∈L∞(0,T;H2(R3)),(u,b)∈ L2(0,T;H13/4(R3))and ω∈L2(0,T;H5/2(R3)).