Analysis Of The Behavior For A Class Of MHD Equations With Hall Term

This paper mainly studies a class of MHD equations with Hall term:Hall-MHD e-quation and its generalized version,generalized two-fluid MHD model.The Hall term is believed to be an essential feature in magnetic reconnection which is happening in the case of large magnetic shear and is useful in describing many physical phenomena in geophysics,astrophysics and plasma physics.This article is devoted to discusses the well-posedness and long time behavior of these equations,with some finite time blow-up criterion results.Firstly,we consider the Cauchy problem of the viscous,resistivity incompressible Hall-MHD equationsBy using Holder inequality,algebraic properties of valuation spaces together with Young in-equality,we prove that the Cauchy problem of Hall-MHD equations is local well-posedness with initial data in low regularity spaces Hs(R3)(3/2<S≤2).While dealing the Hall term with strong nonlinearity,commutator estimates and Sobolev embeddings play a key role in the reduction of the index.In addition,we obtain the global existence of solutions for the small initial data.Secondly,for the viscous,resistivity incompressible generalized Hall-MHD systemwe consider the global existence of small initial data solutions.While dealing with the high-er nonlinear term in the estimate of higher regularity in magnetic field,firstly transferring the first order derivatives in convection term by making full use of integration by parts,then by Kato-Ponce commutator estimates and Sobolev embeddings,we enlarge the range of dissipative exponents α,β from α=β∈(1,7/6]to α＝β∈(1,3/2).Meanwhile,the long time behavior of the corresponding solution for α=β∈[1,5/4)is studied.subsequently,the blow-up criterion of 3D generalized Hall-MHD systems for the general initial data is studied.By means of the techniques of Fourier localization,Bony product decomposition,Sobolev embeddings,interpolation and Young inequalities,we obtain the blow up criteria in a more general Besov space.Finally,we are focused on the incompressible generalized two-fluid MHD systemFirstly,by making full use of commutator estimates,Sobolev embeddings,interpolation and Young inequalities,for α=β∈(1,3/2),we prove the existence of local in time solution in the setting of Sobolev spaces Hm(R3)×Hm+1(R3),m>7/2-2α for the Cauchy problem of generalized two-fluid MHD system.Secondly,using Fourier localization technique and commutator estimates,we obtain the regularity criterion of local solutions at t = T moment.