Optimal regularity in time and space for stochastic porous medium equations

TL;DR

This paper establishes optimal regularity estimates in Sobolev spaces for solutions to stochastic porous medium equations with multiplicative noise, extending deterministic results to stochastic settings using novel velocity averaging techniques.

Contribution

It introduces a new mixed kinetic/mild solution representation and adapts velocity averaging techniques to the stochastic $L^2$ context, achieving optimal regularity results.

Findings

Optimal Sobolev regularity estimates for stochastic porous medium equations.

Extension of deterministic regularity results to stochastic equations with multiplicative noise.

Development of a new solution representation and velocity averaging adaptation for stochastic PDEs.

Abstract

We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be H\"older continuous and the cases of smooth coefficients of at most linear growth as well as $\sqrt{u}$ are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [Gess 2020] and [Gess, Sauer, Tadmor 2020] and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual $L^1$ context to the natural $L^2$ setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use $L^2$ based a priori bounds to treat the stochastic term.