H 1 -Random Attractors and Asymptotic Smoothing Effect of Solutions for Stochastic Boussinesq Equations with Fluctuating Dynamical Boundary Conditions

Abstract

This work is concerned with the random dynamics of two-dimensional stochastic Boussinesq system with dynamical boundary condition. The white noises affect the system through a dynamical boundary condition. Using a method based on the theory of omega-limit compactness of a random dynamical system, we prove that the $L^2$-random attractor for the generated random dynamical system is exactly the $H^1$-random attractor. This improves a recent conclusion derived by Brune et al. on the existence of the $L^2$-random attractor for the same system.