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

DOI10.1214/13-AOP869zbMATH Open1385.37082arXiv1108.2413OpenAlexW3106022634MaRDI QIDQ5920324

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Publication date: 25 April 2014

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Abstract: Unique existence of solutions to porous media equations driven by continuous linear multiplicative space-time rough signals is proven for initial data in $L^1(mathcalO)$ on bounded domains $mathcalO$. The generation of a continuous, order-preserving random dynamical system on $L^1(mathcalO)$ 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 $L^{infty}(mathcalO)$ 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 $L^{m+1}(mathcalO)$.

Full work available at URL: https://arxiv.org/abs/1108.2413

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