Global Well-posedness Of The Two Kinds Of Fluid Mechanics Equations

In this paper,we study global well-posedness of the two-dimensional incompressible Magneto-Micropolar equations with partial dissipation and a generalized surface quasi-geostrophic equation,The details are as follows:Consider two-dimensional magneto-micropolar equations:where x∈R2,t>0,u=u(x,t),b=b(x,t),ω=ω(x,t)and p=p(x,t)denote the velocity of the fluid,the magnetic and micro-rotational velocity,μ,χ and 1/v(v>0)are,respectively,kinematic viscosity,vortex vicosity and magnetic Reynolds number,k is angular viscosities.The global existence and uniqueness of the solution was obtained when it full dis-sipation.However,in the case of inviscid Magnetic-micropolar equations,the global regularity problem is still a challenging open problem.Therefore,it is natural to study the intermediate cases with partial dissipation,namely,we remove some directional dissipations in the velocitv YJy the magnetic b and micro-rotational velocity to.We ap-ply anisotropic Sobolev inequality to overcome the difficulties caused by the lack of dissipation.Then we obtain the global regularity of solutions under the two kind of dissipation conditions:Assume(u0,ω0,b0)∈H2(R2),and ▽·u0=▽·b0=0,then two-dimensional Magneto-micropolar equations(0.0.3)have a unique global classical solution(u,ω,b)∈ L∞°([0,∞);H2(R2)).Consider the dissipative active scalar equations:where k>0,β∈(0,1]and α≥0,θ is a scalar function and ∧=(-△)1/2 and ∧rf/(ξ)=|ξ|rf(ξ)for(r>0).Because the equation has global regularity in the critical and subcritical conditions,we turn to study a class of generalized surface quasi-geostrophic equation with the supercritical condition.Firstly,we establish a regularity criteria θ∈L∞([to.t];Cδ(R2)),and we apply it and nonlinear maximum principle to prove eventual regularity.Then we use the method of energy estimation to prove that the equation(0.0.4)has a unique local classical solution and obtain(?)where(C1>0).Lastly,we obtain that existing α0 =α0(R)∈(0,β)such that for every α∈[α0,β],the equation(0.0.4)has a unique classical global solution,when θ0 ∈H2(R2)and(?).