Global Well-posedness Of The Incompressible Generalized Hall-magnetohydrodynamics Equations

In this dissertation,we investigate the 3D incompressible generalized Hall-magnetohy drodynamics equations:(?)where t?0,x G ?R3,u(x,t)and b(x,t)denote the velocity field and magnetic field of the fluid respectively,p(x,t)is the fluid pressure,v and ? are the viscosity coefficient and magnetic Reynolds number,respectively,and u(x,0)= u0(x),b(x,0)= b0(x)are the initial value of the velocity field and magnetic field.The Laplacian—? in the dissipation terms have been replaced by general multiplier operators with symbols given by m1 and m2,namely ?1u(?)=m1(?)u(?)and ?2b(?)=m2(?),(?).When ?12u =-?u,?22b=-?b,(0.0.4)becomes the incompressible standard Hall-magnetohydrodynamics(Hall-MHD)equations.Hall-MHD system was derived strictly from either Euler-Maxwell equations or kinetic model.Note that if ? ×((?×b)×b)is removed,the Hall-MHD equations become the magnetohydrodynamic equations(MHD).And if b = 0,it reduces to the famous Navier-Stokes equations.For the three dimensional incompressible standard Hall-MHD equations,the regu-larity of weak solutions is still unsolved.The major difficulty is that the dissipation of-fered by the Laplacian operator is insufficient to control the nonlinearities.It is necessary to study it as an important topic in the research of global well-posedness.Wan obtained the global smooth solutions for the three dimensional incompressible Hall-MHD equa-tions with the critical and supercritical hyperdissipation m12(?)=|?|2?,m22(?)=|?|2?,and ??5/4,??7/4 in 2015.In this dissertation,we improve Wan's result by making two different type of reduction in the dissipation,and obtain the global regularity of weak solutions and the existence and uniqueness of weak solutions in H1-functional setting,respectively.In the third chapter,we research the regularity of weak solutions to the three di-mensional incompressible Hall-MHD equations(0.0.4).Inspired by bringing in the log-arithmic factor in the dissipation to the generalized Navier-Stokes equations and MHD equations,we make a logarithmic reduction in the dissipation to the three dimensional incompressible generalized Hall-MHD equations,and obtain a unique smooth global so-lutions by the Littlewood-Paley decomposition.More precisely,(u0,b0)? HS(R3)and s>5/2,the dissipations are m1(?)?(|?|?)/(g1(?),and m2(?)?(|?|?)/(g2(?)for all sufficiently largge ? and some non-decreasing functions g1 and g2:R+?R+ satisfying ?1?(ds)/(s(g12(s)g22(s))2=+?,where ??5/4,??7/4,and we obtain that the generalized Hall-MHD equations have a unique smooth solutions.In the fourth chapter,we study the three dimensional incompressible fractional Hall-MHD equations with partial directional dissipation,whose dissipation terms are:m1|?|=(|?1|5/4+|?2|5/4+|?3|5/4,|?|5/4+|?3|5/4),m2|?|=(|?1|7/4+|?2|7/4+|?|7/4,|?|7/4).(0.0.5)We remove some directional dissipations in the hyperdissipation of velocity field and magnetic diffusion,and we obtain the global existence and uniqueness of weak solutions in the H1-functional setting through the energy estimate.In particular,our result is:assume u0,b0?H1(R3)with ?·u0=?·b0=0,then the equations(0.0.4)with the dissipation terms(0.0.5)have a unique global solutions(u,b)in the sense of H1,and the H1-norm of a and b is bounded uniformly about time.In order to see clearly the structure of the Hall term,we use the least directional dissipation to control the nonlinearities in the process of energy estimate.The dissipa?tion in(0.0.5)is put to use for the estimates of the three difficult terms in the Hall term.As regards other terms,we use the partial directional dissipation in(0.0.6)as follows m1|?|=(|?1|5/4+|?2|5/4+|?3|5/4,|?1|5/4+|?3|5/4),m2|?|=(|?1|7/4+|?2|7/4+|?3|7/4,|?1|7/4+|?3|7/4),(0.0.6)When the Hall term is omitted in system(0.0.4),it becomes the three dimensional incompressible generalized magnetohydrodynamic(MHD)equations.As a deduction,we obtain the existence and uniqueness of the global solutions(u,b)in the H1-functional setting for the generalized MHD equations,and the H1-norm of a and b is bounded uniformly about time if the dissipation terms are(0.0.6).