Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise
- Authors: Benjamin Gess
- Preprint year: 2011
- First public date: 2011-08-11
- arXiv: 1108.2413
- Status: Published
- Publication type: Journal article
- Publication year: 2014
- Journal: The annals of probability, 42 (2014) 2, pp. 818-864
- DOI: 10.1214/13-AOP869
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 $L^{m+1}(\mathcal {O})$.