Smooth Approximation And Inertial Form For A Class Of Stochastic Partial Differential Equations

This work mainly provides the inertial form and smooth approximation for a class of stochastic partial differential equations with white noise,taking into account its additive noise and multiplicative noise,respectively.The infinite dimension of the state space makes the system too complicated to be visualized geometrically and too difficult to be analyzed clearly.Using the method of invariant random cone,the solution of original system can be reduced to the finite dimensional invariant manifold with an exponential convergence rate.Furthermore,the infinite dimensional system can be reduced to the inertial form of finite dimensions.In addition,since that the white noise is generated by Brownian motion which is continuous everywhere but non-differential everywhere,this thesis adopts a new stochastic evolution equation with smooth colored noise to approximate the original equation.Further,it is proved that when the smooth colored noise tends to the white noise,the solution and the inertial form of the new approximate system converge to those of the original system.