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 $L^1(\mathcal{O})$ 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 $L^{m+1}(\mathcal{O})$.

Published in at http://dx.doi.org/10.1214/13-AOP869 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Random attractors for a class of stochastic partial differential equations driven by general additive noise
• Rough evolution equations
• Random attractors for degenerate stochastic partial differential equations
• Stochastic Porous Media Equation and Self-Organized Criticality

Cited by in corpus (6)

• Finite time extinction for stochastic sign fast diffusion and self-organized criticality
• Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations
• Distribution Dependent Stochastic Porous Media Equations
• Supremum estimates for degenerate, quasilinear stochastic partial differential equations
• Optimal regularity in time and space for stochastic porous medium equations
• Expansion and attraction of RDS: long time behavior of the solution to singular SDE