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

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

Strong Solutions for Stochastic Partial Differential Equations of Gradient Type

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the "distance" defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.Comment: 30 page

Stochastic continuity equations with conservative noise

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence and uniqueness of entropy solutions under hypotheses on the velocity field which are weaker than those required in the deterministic setting. This extends related results of [Flandoli, Gubinelli, Priola; Invent. Math., 2010] applicable for linear multiplicative noise to a non-linear setting. The existence proof relies on a duality argument which makes use of the regularity theory for fully non-linear parabolic equations.Comment: 42 page

The stochastic thin-film equation: existence of nonnegative martingale solutions

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise in one space dimension and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous work, no interface potential has to be included, the initial data and the solution can have de-wetted regions of positive measure, and the Trotter-Kato scheme allows for a simpler proof of existence than in case of It\^o noise.Comment: 38 pages, revised version, nonnegativity proof changed, details to time regularity and interpolation of operators adde