Global Well-posedness Of The Incompressible Magnetohydrodynamics Equations

In this thesis, we investigate the generalized incompressible magnetohydynamics system in Rn(n=2,3). where u(t,x),b(t,x),p are the velocity field, the magnetic field and the pressure, re-spectively, v is the viscosity coefficient, which is equal to the reciprocal of the Reynold number, and ?is the electrical resistivity coefficient, which is inversely proportional to the electrical conductivity constant, and ?? 0 and ?> 0 are real parameters, A= (-?)2 is defined in terms of Fourier transform by Af(?)=|?|f(?). If ?=?=1, the system (0.0.3) reduces to the famous incompressible MHD system: This system systematically describes the macroscopic behavior of incompressible fluid conduction system. The MHD system simulates many phenomena in our daily life. For example, the phenomenon of magnetic power generation in geophysics, the solar wind and the sunlight reflection phenomenon in astrophysics [29]. Especially, if b= 0, the MHD system reduces to the classical incompressible Navier-Stokes equation.In the third chapter, we study the global regularity of the solutions to the 2D gener-alized MHD system (0.0.3). By virtue of the energy method and some basic inequalities and the properties of the Sobolev space, we study the GMHD equation's regularity cri-terion. Assume (uo,bo) ? Hs with s> 2. Then the system (0.0.3) is globally regular for ? and ? satisfying 0<???1/2,?+?>3/2.In the fourth chapter, we research the asymptotic stability of the mild solution to the 3D incompressible MHD system (0.0.4). It is well-known that the global-in-time solution with the small initial data to the system (0.0.4) was obtained in reference [31]. Here we denote (V(x,t),B(x,t)) a global-in-time solution, we give a L2 perturbation to the solution of the system (0.0.4), then we get a perturbed equation.Let (V = V(x, t), B = B(x, t)) be a global-in-time solution to the small initial value problem (0.0.4) with ||V0||H1/2 ||B0||H1/2?min{?0,1}- Denote V0 = V(·,0),B0 = B(·,0) and let (w0,h0) ? L2(R3) be arbitrary. Then, the Cauchy problem (0.0.4) with the initial condition u0 = V0 + w0, b0 = B0 + h0 has a global weak solution (u, b) of the form u(x,t) = V(x,t) + w(x,t),b(x,t) = B(x,t) + h(x,t), where (w(x,t),h(x,t)) is a weak solution of the corresponding perturbed problem satisfying (w, h) ? Cw([0, T]; L2(R3)) (?) L2([0,T];H1(R3)) for each T > 0.As t? ?, the weak solution (u(x,t),b(x,t)) of the system (0.0.4) tends to the mild solution (V = V(x,t),B = B(x,t)) in sense of the L2—norm, namely ||w(t)||2 ||u(t) - V(t)||2 ?0, ||ht)||2 = ||b(t) - B(t)||2 ?0, where (w(x,t),h(x,t)) is the weak solution of the perturbed equation.