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Optimal regularity in time and space for stochastic porous medium equations. (English) Zbl 1500.35072

Summary: 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 \(\sqrt{u}\) are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [B. Gess, J. Eur. Math. Soc. (JEMS) 23, No. 2, 425–465 (2021; Zbl 1461.35133); B. Gess et al., Anal. PDE 13, No. 8, 2441–2480 (2020; Zbl 1459.35254)] 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.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K59 Quasilinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76S05 Flows in porous media; filtration; seepage
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