Optimal regularity in time and space for stochastic porous medium equations
A talk in the Bielefeld Stochastic Afternoon series by
Stefano Bruno
| Abstract: | This talk will be held via Zoom (for details email stochana(at)math.uni-bielefeld.de). Stochastic porous media equations are well-studied models describing non-linear diffusion dynamics perturbed by noise. In this talk, we consider the noise to be multiplicative, white in time and coloured in space and the diffusion coefficients to be Hölder continuous. Our assumptions include the cases of smooth bounded coefficients as well as $\sqrt{u}$ -coefficients relevant in population dynamics. Using the kinetic solution theory for conservation laws, we prove optimal regularity estimates 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 of our result 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. In particular, we introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDEs and use new $L^2$-based a priori bounds to treat the stochastic term. Within the CRC this talk is associated to the project(s): B8 |