Existence And Large Time Behavier Of Solution To The Incompressible Navier-Stokes-Landau-Lifshitz-Bloch Equation

Magnetohydrodynamics is widely used in astrophysics,controlled thermonuclear and industrial fields.In order to futher grasp the characteristics of magnetohydrodynamics,it is necessary to carry out theoretical research on the mathematical models related to magnetohydrodynamics.In this thesis,we mainly study the existence and large time behavier of solution to the incompressible Navier-StokesLandau-Lifshitz-Bloch equation.In this thesis,we first prove the global existence of weak solution to equation in two-dimensional and three-dimensional periodic domains by using classical Faedo-Galerkin method and compachtness theory.Secondly,the existence of smooth solution in three-dimensional space under the condition of small initial value is proved by energy method,and then the decay rate of the solution with time and decay rate of spatial higher-order derivatives of the smooth solution are obtained by a Fourier splitting method,which are ‖u‖L²（R³）+‖d‖L²（R³）≤C（1+t）^-3/4 and ‖D^mu‖L^p（R³）²+‖D^md‖_L^p（R³）²≤C_m（t+1）^{（-m-3）（1/2-1/p）-2q},respectively.Finally,on this basis,it is obtained that the decay rate for the mixed space-time derivatives of the solution is ‖D^l?_tu‖_L²（R³）²+‖D^l?_td‖_L2（R3）²≤C（t+1）^-（l+3/2）.