Large Time Behavior Of Solutions Of MHD And NS Equations With Partial Viscosity

The mathematical models of fluid dynamics are important models in the theory of PDEs and described by momentum,the mass conservation and energy conservation to-gether with the basic laws of thermodynamics.They play an important role in theoretical and applied studies in atmospheric and oceanographic sciences and petroleum industries.The mathematical theory in fluid dynamics has been improving since French mathemati-cian Jean Leray gave the existence of weak solutions of viscous incompressible fluid flows in 1930’s.The fundamental issues which have longstanding interest to mathematicians are the well-posedness and large time behavior of the solutions.This paper is mainly devoted to studying the behaviors of one special classical equations modelling fluid dynamics with partial viscosity.More precisely,we are focused on the regularity and decay of solutions to the special nonlinear partial differential equations in MHD,Navier-Stokes and isotropic non-Newtonian flows.The main results and proof are made in the chapter 2-chapter 4.In Chapter 1,we first introduce three types equations,some notations and basic inequalities.In Chapter 2,we consider the Cauchy problem of two-dimensional magnetohydrody-namic equations with partial viscosity,including the global existence and decay of higher order norms of smooth solutions.In one special group of partial viscosity,we obtain the global existence of the smooth solution by taking advantage of the estimate of energy,giving the estimate of L2,H1 and Hs（s ≥ 2）and the L2 decay of smooth solutions and Hs（s ≥ 2）decay which is decay of higher-order derivatives,by making use of the classic technology of Fourier splitting and the mathematical induction.In Chapter 3,we focus on the optimal algebraic decay of solutions for two-dimensional Navier-Stokes equation with partial viscosity.By developing the classic Fourier Splitting methods together with inductive methods,the higher-order derivatives of solutions are proved to decay in the optimal algebraic rates ‖▽^s（t）‖L²≤C（1+t）^-（s+1）/2,s≥0,t＞0.In Chapter 4,we focus on the L2 decay of weak solutions of the isotropic non-Newtonian flows with partial viscosity in whole space R2.By using the energy methods and modified Fourier splitting methods,we proved the optimal time decay rates of weak solutions for the system as ‖u‖L²（R²）≤C（1+t）^-（1/2）.