The Existence And Uniqueness For The Stochastic Navier-Stokes Equations

The study of stochastic partial differential equations is an important branch of stochastic analysis,and is widely used in many fields such as natural science and eco-nomics.There exists a great amount of literature on the stochastic differential equations with Gaussian noise.However,in financial,physical and biological phenomena,Gaus-sian noise has many imperfections while capturing some discontinuous events.Since Levy noises can describe a large class of discontinuous phenomena,considering the stochastic model driven by Levy noises becomes extremely popular.This thesis takes the 2-D stochastic Navier-Stokes equation(SNSE)as an example,the main aim is to summarize the method of studying the existence and uniqueness of solutions to stochastic partial differential equations driven by Levy noise.The structure and specific content of this thesis are as follows:Chapter 1 briefly introduces the significance and current situation of stochastic differential equations with Levy noise.Chapter 2 mainly give some fundamental knowledge related to this thesis.In Chapter 3,martingale solutions of the 2-D SNSE driven by the Levy type noise are considered.Using the classical Faedo-Galerkin approximation,Aldous condition and a version of the Skorokhod Embedding Theorem for nonmetric spaces method,the existence of a martingale solution is obtained.This method is also applicable to the 3-D cases with bounded and unbounded domains.Chapter 4 discusses the existence and uniqueness of the strong solution of the 2-D SNSE in probabilistic sense by the classical Galerkin approximation and the local monotone method.This method is also applicable to other abstract nonlinear equations with Levy noise.In Chapter 5,using the cut-off and Banach Fixed Point Theorem,the existence and uniqueness of strong solutions for the 2-D SNSE with Levy noise in PDE sense are obtained.This method can also prove the existence and uniqueness of strong solutions in probabilistic sense,and only need coefficients satisfy Lipschitz and linear growth condition,but the method in chapter 4 requires other assumptions.In Chapter 6,applying exponential transform and Malliavin analysis,we consider the existence and uniqueness of the solution for the SNSE driven by Wiener noise and with random initial conditions.Chapter 7 summarizes the methods introduced in this thesis.