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

TL;DR

This paper proves the existence of solutions, random attractors, and their properties for stochastic porous media equations driven by space-time rough signals, including fractional Brownian motion, on bounded domains.

Contribution

It establishes the generation of a continuous, order-preserving random dynamical system and the existence of a compact random attractor for such equations with general noise.

Findings

Existence of solutions for initial data in L^1 and L^{m+1} spaces.

Construction of a continuous, order-preserving random dynamical system.

Existence of a compact, attracting random attractor in L^{} norm.

Abstract

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space-time rough signals is proven for initial data in $L^1(\mathcal {O})$ on bounded domains $\mathcal {O}$. The generation of a continuous, order-preserving random dynamical system on $L^1(\mathcal {O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^{\infty}(\mathcal {O})$ norm. Uniform $L^{\infty}$ bounds and uniform space-time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated and a pathwise Wong-Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space-time rough signals, existence of solutions is proven for initial data in $L^{m+1}(\mathcal {O})$.