Attractors For Delay Kuramoto-Sivanshinsky Equation And Its Approximation

This essay mainly studies the long-term behaviour of delay Kuramoto-Sivanshinsky equation with multiplicative noise on a bounded domain I=[-l/2,l/2]as follows.Firstly,we introduce the background and current research status of random dynamical system,random attractors and stochastic KS equation.Then,we recall the definition to random pullback absorbing set,asymptotic compactness of random dynamical system,and particularly present the theorem which guarantees the existence of random attractors.As an example of studying stochastic differential equation by random dynamical system,we prove,as follows,the existence of attractor to the differential equation above.Next,we give the random dynamical system determined by the differential equation.This is based on the existence and uniqueness of the solution to that equation.Precisely,we transfer the stochastic differential equation with multiplica-tive noise into a deterministic equation by an exponential transformation.Then the existence and uniqueness to the latter equation can be acquired by Galerkin's method.That is,the latter equation has a unique solution on C([-?,0],L~2(I)).By inverse transformation of the former exponential transformation,the existence and uniqueness of solution to the original differential equation is obtained,and a continuous random dynamical system is defined based on this,that is,a family of operators popes out by the solution to the equation which can be proved to be a continuous random dynamical system.This step we prove the existence of a random absorbing set.According to its definition,this is equivalent to do some uniform estimate on C([-?,0],L~2(I)).In fact,by taking inner product on both sides of the deterministic equation,we could get the expression of the solution on C([-?,0],L~2(I)).Together with the restriction of the nonlinear term and the delay of the initial equation,we can prove the absorption to the solution.To prove asymptotic compactness of the random dynamical system,we need uniform estimate on space with higher regularity,i.e.on C([-?,0],H01(I).This is due to the fact that(C([-?,0],H01(I))compactly embedding into C([-?,0],L~2(I)).The theorem to the existence of the random attractors guarantees the stochas-tic delay KS equation with multiplicative noise has a unique random attractor.Furthermore,we consider periodicity and upper semi-continuity of the attractor.