Global Well-posedness Of The Incompressible Hall-magnetohydrodynamics Equations

In this paper,we study the 3D incompressible Hall-magnetohydrodynamics equa-tions:where u =(u(x,t)and B = B(x,t))denote the velocity field and magnetic field of the fluid,respectively,p = p(x,t)is the pressure,v and ? is the viscosity coefficient and electrical resistivity coefficient,respectively.Acheritogaray,Degond,Frouvelle and Liu[1]derived Hall-MHD from two-fluids or kinetic models.Note that if B= 0,the Hall-MHD becomes Navier-Stokes equation.And if ?×((?×B)×B)= 0,it reduces to the magnetohydrodynamic equations.Firstly,the global existence of the weak solution of the three dimensional incom-pressible Hall-MHD equations is proved by using the Friedrich method in our paper:if u0,B0?L2(R3)and ?·u0=?·B0= 0,then the Hall-MHD equations have global weak solutions(u,B).Secondly,we get the regularity criterion in terms of one direc-tional derivative of velocity field(?)iu and the gradient of magnetic field ?B.Thirdly,we establish the blow-up criterion of the smooth solutions to the three dimensional incom-pressible resistive viscous Hall-magnetohydrodynamic equations in terms of u and ?B or ?u and ?B in Lebesgue spaces or ?u in Besov space B?,?0 and ?B in BMO space,where u denotes(u1,u2).We also obtain a priori estimate of velocity and magnetic field in the space of H3/2(R3)on[0,T].