Existence And Periodicity Of Solutions For Some Stochastic Partial Differential Equations

Stochastic partial diferential equations are used to model many physical systemssubjected to the influence of internal, external or environmental noise. In the thesis,we study the existence and periodicity of some stochastic partial diferential equations.In Chapter2, we study the almost automorphic solutions of stochastic diferen-tial equations driven by fractional Brownian motion. We study two cases of Hurstparameter taking values in the interval (12,1) and (0,12). We obtain the existence ofalmost automorphic solutions in distribution, and can not get mean square almostautomorphic solutions. In the end of this chapter, we give some counterexamples ofmean square almost automorphic solutions for stochastic diferential equations, and aexample is given to illustrate the existence of almost automorphic solutions for heatequations in distribution.In Chapter3, we study stochastic fractional Burgers type nonlinear equations driv-en by fractional double-parameter noise. Our equation contains a pesudo-diferentialoperator which is a Markovian generator of a stable-like Feller process with variableorder α(x). Under some conditions, we establish the existence and uniqueness of a localsolution for the initial value problem. Furthermore, we obtain the joint continuity ofsolutions for the stochastic fractional Burgers equationsIn Chapter4, we study a class of stochastic Cahn-Hilliard equations driven bymulti-parameter fractional Brownian motion. We provide the existence and uniquenessof the solution for these equations, and use Malliavin calculus to prove that the law ofthe solution is absolutely continuous with respect to Lebesgue measure onRd. In Chapter5, we study a rigid body-fluid system. There is a rigid body with finitevolume and arbitrary shape moving in unbounded incompressible fluid. We assume thatthe fluid motion continuously, starting from static condition, which is forced by themotion of rigid body. We also assume that the fluid at infinity is static. Here the fluidis subject to a time-periodic body force and an additional time-periodic stochastic forcethat is produced by a rigid body moves periodically stochastically with the same periodin the fluid. These physical phenomenon can be modeled by the stochastic Navier-Stokes equations around a moving body. The existence of time-periodic stochasticsolution of the equations is obtained.