Dynamics Of The Stochastic Fractional Partial Differential Equations

This thesis is devoted to the dynamics containing the well-posedness,the existence of a random attractor and ergodicity of the stochastic fractional hydrodynamical-type equation including the fractional Boussinesq equation,coupled fractional Ginzburg-Landau equation,fractional MHD equation,hydrodynamical-type evolution equation,GinzburgLandau-Newell equation and two-dimensional primitive equations of large scale ocean in geophysics driven by Gaussian white noise,Lévy noise,?-stable noise and degenerate noise.Finally,we show the well-posedness of the stochastic time-space fractional Ginzburg-Landau equation and Boussinesq equation driven by Gaussian white noise.The dissertation consists of five chapters.In Chapter 1,we first introduce the physical background and latest advanced in the fractional differential equations and the infinite-dimension dynamical system.Then we present the basic concepts,formulas and inequalities which the following chapters depend on.Then we state our main results.In Chapter 2,the fractional commutator estimates and fractional Sobolev embedding theorem are used to over the difficulties that the higher order estimates cannot be obtained.The global existence and uniqueness of the weak solution for the stochastic fractional Boussinesq equation with Lévy noise are obtained in a fixed probability space.Then we prove the existence of a random attractor for the stochastic coupled fractional GinzburgLandau equation driven by Gaussian white noise,the stochastic fractional Boussinesq equation driven by Gaussian white noise and the stochastic Fractional MHD equation driven by Gaussian white noise.The conditions of the fractional operator are given.In Chapter 3,we introduce the bstract stochastic hydrodynamical-type evolution equation driven by ?-stable noise which includes the Navier-Stokes equation,Boussinesq equation,MHD equation,magnetic Benard equation.By the Banach fixed point theorem,we prove the existence and uniqueness of the mild solution.Then we show the strong Feller and the accessibility to get the uniqueness of the invariant measure.Also we use the model to get the ergodicity of the Boussinesq equation and MHD equation.In addition,the ergodicity of the stochastic coupled fractional Ginzburg-Landau equation driven by ?-stable noise is shown by the same method.In Chapter 4,we use the It?'s formula and stopping time technique to show the momentum estimates,the existence and uniqueness of the martingale solutions for the stochastic Ginzburg-Landau-Newell equation driven by degenerate noise,the stochastic fractional Boussinesq equation driven by degenerate noise,the stochastic fractional MHD equation driven by degenerate multiplicative noise,the stochastic two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise.Then we prove the asymptotically strong Feller property and the support property to show the uniqueness of the invariant measure.In addition,for the stochastic fractional MHD equation driven by degenerate multiplicative noise,we get the irreducibility and the asymptotically strong Feller property to show the uniqueness of the invariant measure.In Chapter 5,we study the stochastic Time-Space fractional Ginzburg-Landau equation driven by Gaussian white noise and the stochastic Time-Space fractional Boussinesq equation driven by Gaussian white noise.The time-spatial regularity of the nonlocal stochastic convolution in one dimension and two dimension are shown.The restriction are imposed on the order of fractional derivative and the order of spatial nonlocal effects.Finally,we prove the existence and uniqueness of the mild solutions by the the Banach fixed point Theorem.