Properties Of Solutions To The 3D Incompressible Magnetohydrodynamics Equations

In this dissertation,on the one hand,we are committed to study the global wellposedness,analytic and decay estimates of small initial data to the 3D incompressible magnetohydrodynamic equations and related models in some critical spaces.On the other hand,we consider some Liouville theorems of the 3D stationary incompressible magnetohydrodynamics equations and related models in L^p((?)³).Firstly,we study the temporal decay estimate of small global solutions to the 3D generalized incompressible magnetohydrodynamic equations in the critical space ?^1-2?((?)³)which describe the motion of conductive fluid under the action of magnetic field,where u =(u₁,u₂,u₃),b =(b₁,b₂,b₃)and ? denote the velocity field,magnetic field and scalar pressure of fluid,respectively.The positive constants ? and (1/?) are the viscosity coefficient and magnetic Reynolds number.The fractional Laplacian operator ?^2?=(-?)^? is defined through a Fourier transform,namelyIn chapter three,we prove two important results.First,for the 3D generalized incompressible magnetohydrodynamic equations in a critical space ?^1-2?((?)³) with (1/2)? ? ? 1,we obtain the temporal decay rate:for a global solution with small initial data which was established by Ye Zhuan.Our result improves the temporal decay rate.Second,for the 3D generalized incompressible magnetohydrodynamic equations near equilibriumWe establish the global well-posedness and analytic estimates for small perturbation by the semigroup method in the critical space ?^1-2?((?)³) with (1/2)? ? ? 1,where linear terms from perturbation incur much difficulty and by introducing a diagonalization process we successfully eliminate the linear terms.Furthermore,we prove the time decay estimate?(u,b)(t)??^1-2?(?)(1 + t)^-(5/(4?)-1) by using the result of the analyticity of the solution.In chapter four we study the global well-posedness and large time behavior to the 3D incompressible magnetohydrodynamic equations(0.0.3)(for ? = 1)in critical spaces(?)^1/2((?)³)and L³((?)^(?)3,respectively.First,we prove a priori estimates by the semigroup method,and derive the global well-posedness and analytic estimate for a solution with small initial data in combination with the bootstrap argument(continuous mathematical induction)in the critical space(?)^1/2((?)³),and we obtain that,for (?)₀> 0 small enough,(u₀,b₀)?(?)^1/2((?)³),and ?(u₀,b₀)?(?)^1/2< (?)0,the equations admit a unique global solutionfor all t > 0.Second,we derive the decay rate ?(u,b)(t)?(?)^1/2(?)(1 + t)^-(1/4) by applying the analyticity and the properties of a continuous function.Last,by means of the standard energy estimate method,we derive that the solution ?(u,b)(t)?L³ of magnetohydrodynamic equations is a Lyapunov function with respect to time and lim_t???(u,b)(t)?L³ = 0.In chapter five,we study some Liouville theorems of the 3D stationary incompressible Navier-Stokes,Hall-magnetohydrodynamics and magnetohydrodynamics equations.By selecting the appropriate cut-off function,the method of H?lder inequality and energy estimation used intricately,we obtain that,if a smooth solution u ? L^p((?)³),2 ? p <(18/5) to the 3D stationary incompressible Navier-Stokes equation,then u = 0.If a smooth solution of 3D stationary incompressible Hall-magnetohydrodynamics equations(u,b)?L^p((?)³),2 ? p ?(9/2),?b ? L²((?)³),or u ? L^p((?)³),2 ? p ?(9/2);b ? L^p((?)³),4 ? q ?(9/2),then u = b = 0.If a smooth solution of 3D stationary incompressible magnetohydrodynamics equations(u,b)?L^p((?)³),2 ? p ?(9/2),then u = b = 0.