Optimal regularity in time and space for stochastic porous medium equations

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 Hölder continuous, and the cases of smooth coefficients of, at most, linear growth as well as $\sqrt{\mathit{u}}$ are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in (J. Eur. Math. Soc. 23 (2021) 425–465, Anal. PDE 13 (2020) 2441–2480) and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual ${\mathit{L}}^1$ context to the natural ${\mathit{L}}^2$ setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use ${\mathit{L}}^2$ based a priori bounds to treat the stochastic term.