Finite Dimensional Reducing Of Singular Perturbed SPDEs Under Different Boundary Conditions

This thesis concerns with three kinds of stochastic partial differential equations driven by degenerate noise,which are a stochastic partial differential equation with quadratic nonlinearities,a singular perturbed stochastic reaction-diffusion system with random Neumann boundary conditions,and a stochastic partial differential equation under fast dynamical boundary conditions.The main purpose is to derive their effective approximating systems.More precisely,it is firstly concerned with a class of stochastic partial differential equations with fast random dynamical boundary conditions.One can transform into an equivalent simpler system with homogeneous boundary conditions by defining corresponding product spaces.In the limit of fast diffusion,it derives one effective stochastic partial differential equation,describing the evolution of the dominant pattern.With the help of multiscale analysis and averaging principle,it establishes deviation estimates of the original stochastic system towards the effective approximating system.Then,it is focused on a class of singular perturbed stochastic reaction-diffusion systems with random Neumann boundary conditions,applied in many fields.It aims to eliminate the disturbing of random Neumann boundary conditions via Neumann mapping,and derive the effective approximating equation of the system.Applying the multiscale analysis and averaging argument,it is shown that the original system converges to a stochastic ordinary equation with some rate as the singular perturbation parameter tends to zero.Finally,it considers a class of stochastic partial differential equations driven by additive degenerate noise,whose nonlinearity is quadratic.Near the change of stability,it uses multiscale analysis to obtain an effective approximating system driven by the kernel space of the original system's operator,furthermore,shows the convergence rate.At last,it illustrates the main results by means of two simple applications.