Stable Random Attractors For Stochastic Kuramoto-Sivashinsky Equation With Large Viscosity

This paper mainly studies the random dynamics for the stochastic Kuramoto-Sivashinsky equation approximated by multiplicative difference noise.Under the assumption of large viscosity,we prove the existence of random attractors for the Kuramoto-Sivashinsky equations driven by difference noise and white noise,respec-tively.Moreover,we establish the stability of the random attractor when the size of difference noise tends to zero,which is just the upper semi-continuity problem.In Chapter 1,we introduce evolution of stochastic dynamical system and ran-dom attractor,and the background and research status of stochastic Kuramoto-Sivashinsky equation.In Chapter 2,we present some preliminary definitions that will be used in this thesis.In Chapter 3,we study the existence of random attractors for the stochastic Kuramoto-Sivashinsky equation.As an application,we study the following random KS equations with multiplicative white noise and time-dependent forcing:where D=(?),?=(-1/2,1/2)with l>0 and W(·)is a scalar two-sided Wiener process on a probability space(?,F,P).Then,we prove the existence of random attractors for the Kuramoto-Sivashinsky equations driven by difference noise,where the difference noise means a Wong-Zakai process,which is just the difference of the Wiener process.So,we can regard the difference process as approximate noise to model the KS equation:where D=(?),?=(-1/2,1/2)with l>0.In this paper,we have to make an assumption that the viscosity v is suitable large,see Hypothesis I.In this case,the cocycle induced by the KS equation possess a pullback absorbing set.Therefore,we can verify the existence of random attractors for KS equation driven by difference noise and white noise.In Chapter 4,we introduce the definition of upper semi-continuity.Then we prove the convergence of the solution operator when the size of difference noise for the random KS equation approaches zero.Finally,we obtain the stability of the random attractor when the difference noise tends to white noise.