Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Abstract

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in L1(O). The generation of a continuous, order-preserving random dynamical system (RDS) on L1(O) and the existence of a “small” random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. Uniform L∞ bounds and uniform space–time continuity of solutions is shown. General noise including fractional Brownian Motion for all Hurst parameters is contained. A pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained.