Well-posedness And Ergodicity For The 3D Stochastic Fractional Boussinesq Equation

In this paper,we mainly study the well-posedness,large deviation principle and exponential mixing property for a class of three-dimensional stochastic fractional Boussinesq equation.Under the variational framework,we prove that the equation has a unique strong solution when the fractional Laplace operator satisfies certain conditions by using local monotonicity method.And the large deviation principle of stochastic fractional Boussinesq equation is obtained by applying the criterion of large deviation principle for stochastic partial differential equation which satisfies local monotonicity conditions.Finally we prove the exponential mixing property for the stochastic Bousssinesq equation by the method which is used to study the exponential mixing of three-dimensional stochastic Leray- model,and apply the exponential mixing criterion in reference [59] to the stochastic fractional Bousssinesq equation.This paper is mainly divided into the following four parts.In chapter 1,we mainly describe the research background and related research progress of stochastic partial differential equations and fractional order stochastic partial differential equations,and briefly summarize the main contents and main results of this paper.In chapter 2,we briefly introduce some basic knowledge of stochastic partial differential equations and the variational framework for SPDE.In chapter 3,we introduce the three-dimensional stochastic fractional Boussinesq equation model and prove the existence and uniqueness of strong solutions of the equation,and give the large deviation principle satisfied by the equation.In chapter 4,we mainly prove the exponential mixing property of threedimensional stochastic fractional Boussinesq equation.