Ergodicity Of Stochastic Navier-Stokes Equation

Stochastic partial differential equations involve probability theory,partial differ-ential equations,infinite dimensional analysis and other fields.It is not only used to solve practical problems in physics,biology,finance and other fields,but also required to promote the internal development of mathematics.Stochastic Navier-Stokes equa-tions are of great significance to understand,predict and control the dynamic behavior of incompressible fluid and compressible fluid.They are widely used in aerospace,water conservancy,marine engineering and other important fields.In this thesis,we mainly study the long-term behavior of the solutions of stochastic Navier-Stokes equa-tions,and focus on the existence and uniqueness of invariant measures.The study of invariant measures began with the work of Krylov and Bogoliubov,and plays an impor-tant role in the study of Markov process theory.Here,we take stochastic Navier-Stokes equations as examples to introduce the research methods and related results of the exis-tence and uniqueness of invariant measures for stochastic partial differential equations.This thesis is a review article,with main references[1],[8]and[24].On the one hand,we introduce the ergodic results for 2D stochastic Navier-Stokes equations driven by additive white noises and exponential mixing of the 3D stochastic Navier-Stokes equations driven by nondegenerate multiplicative noises and mildly degenerate noises respectively;on the other hand,we take 2D and 3D stochastic Navier-Stokes equations as examples to show how to prove the existence and uniqueness of invariant measures,including by proving the strong Feller property and irreducibility of the Markov semi-group associated with the equation,by using coupling method and malliavin analysis.The thesis is organized as follows.Section 1 is the introduction,which briefly introduces the background knowledge,research methods and development of Navier-Stokes equations and summarizes the main results;Section 2 gives the basic knowledge used in this paper,mainly including the notations of stochastic Navier-Stokes equations,the concepts of invariant measures and coupling method;Section 3 introduces ergodic results for 2D stochastic Navier-Stokes equations driven by white noises,the method introduced in this chapter is to prove the strong Feller and irreducibility of the Markov semigroup associated with this equation;Section 4 proves the exponential mixing of the 3D stochastic Navier-Stokes equations driven by nondegenerate noises,the method introduced in this chapter is Galerkin approximation and coupling method;Section 5 shows the exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises,the method introduced in this chapter is Kolmogorov equation and coupling method.Section 6 summarizes the main results and research methods of this thesis.