Random Attactors Of Kuramoto-Sivashinsky Equation With Multilicative White Noise

The attractor is one of the most important problems recently.And the random at-tractor is that of central parts of the asymptotic dynamics of the stochastic differential equation. This paper is devoted to the existence of the random attractor of stochastic Kuramoto-Sivashinsky equation with multiplicative White Noise in I=(-L/2, L/2):du+(vD4u+D2u+uDu)dt=buo dW(t) (x,t)∈I×RIn chapter 1, this paper introduce the development survey of the stochastic dynamical systems, Documents referenced in this paper and preliminary results and definitions, and frequently used inequalities, then, the author briefly introduce the research work of this paper.In chapter2, we mainly use the transformationα(t)=e-bW(t),v=αu to remove the random items, and then take the Galerkin approximation method to prove the unique solution of equation, the unique solution will generate a RDS. In chapter 3, we consider the existence of the random attractor on the whole space. Since eigenvalues of vD4u+D2u may be negative, the linear operator vD4u+D2u is not coercive.Thus it is difficult(or impossible)to obtain a general result on attractors on the whole space H=L2(I) even for the deterministic case. Therefore, We'll prove the existence of a random attractor which attracts all bounded set of the space H, under the following assumption v/(L2)＞1(4π2) where L is the length of I and v is the given number defined.In chapter 4, We consider the existence of the random attractor on the subspace of odd functions. Here use the introduction of a smooth odd function as an auxiliary function, Using w=v+ψ, we can prove the existence of attractor of w, Then the random dynamical systems, associated with odd solution of the stochas-tic Kuramoto-Sivashinsky equation with multiplicative White Noise possesses a compact random attractor,which attracts all deterministic bounded sets of the subspace H0 of odd functions.