Degenerate Pullback Attractors And Uniform Attractors For Nonlinear Hydrodynamic Equations

The thesis consists of two parts.The first part is the degenerate pullback attractor for the three-dimensional Navier-Stokes equations with nonlinear damping,and the other part is the uniform attractor for the three-dimensional MHD equations with nonlinear damping.The full thesis is divided into the following four chapters:In the first chapter,we mainly introduce the research background of Navier-Stokes equations and MHD equations,and then introduce the research content and the main results of this thesis.In the second chapter,we mainly introduce the basic theoretical knowledge of pullback attractors and uniform attractors.In the third chapter,we study the long time behavior of solutions for the three-dimensional Navier-Stokes equations with nonlinear damping under the condition that the Grashof number is sufficiently small when the external force is translationally bounded in Lloc2(R,L2(Ω)),and then prove the existence of degenerate pullback attractors.To this end,we first prove that a weak solution of the problem(1.1.3)is a Leray-Hopf weak solution and further obtain the existence of pullback attractors.Then we obtain the existence of the strong solution of the problem(1.1.3)by making a critical priori estimation under the condition that the Grashof number is sufficiently small.Finally,based on the uniqueness of the strong solution,we prove that the pullback attractor is degenerate.In the fourth chapter,we study the long time behavior of solutions for the three-dimensional MHD equations with nonlinear damping when the fluid velocity satisfies a no-slip boundary condition and the magnetic field satisfies a time-dependent Dirichlet boundary condition,and then prove the existence of the uniform attractor.To this end,firstly,we introduce lifting functions to homogenize the magnetic field boundary and further obtain energy estimates.Secondly,we prove the existence and uniqueness of global weak solutions.We first prove that the existence of local approximate solutions(um,bm)by using Schauder’s fixed point theorem and semi-Galerkin approximation method,and then prove that the existence of global approximate solutions(um,bm)by making priori estimation.Then,let m→∞ to obtain the existence of weak solutions(u,b)and the uniqueness of weak solutions is obtained by continuous dependence.Thirdly,we prove the existence and uniqueness of global strong solutions.Finally,we prove that the existence of uniform attractors is based on bounded absorbing sets.