The Deviation Theory Of Stochastic Partial Differential Equations Driven By Multiplicative Noise

This thesis is concerned with the deviation theory of three kinds of stochastic partial differential equations driven by multiplicative noise,including stochastic Leray-equation with fractional dissipation,stochastic Kuramoto-Sivashinsky equation and stochastic Cahn-Hilliard-Navier-Stokes equation.Firstly,it focuses on the stochastic Leray- equation with fractional dissipation.Introducing an appropriate state space to eliminate the influence of fractional dissipative term,it establishes the central limit theorem of the system.By building a new approximate system,and combining the stochastic control theory,it further obtains the equivalent form of the approximate system.Applying the weak convergence approach,it derives the moderate deviation principle of the system.Secondly,it considers the stochastic KuramotoSivashinsky equation.Through improving the regularity of Green function,it overcomes the difficulties caused by higher-order nonlinear term.With the help of a new approximate system and the stochastic control theory,it derives the tightness of the control system by constructing an effective control system.Employing the uniform weak convergence approach and a splitting skill,it builds the uniform moderate deviation principle of the system.Finally,it investigates the stochastic Cahn-Hilliard-Navier-Stokes equation.By changing the time increments,it obtains the core structure of the system.Using the stochastic control theory and the weak convergence approach,it derives the moderate deviation principle of the system underlying the verification of two moderate deviation conditions.