Oberseminar AG Analysis - Dr. Sebastian Bechtel, TU Delft, Niederlande

Existence for quasilinear systems of SPDEs in a variational setting

We consider a quasilinear system of SPDEs in a variational setting. We are going to show existence of a so-called martingale solution using a stochastic compactness argument. This approach already proved useful in the literature: Debussche, Hofmanova and Vovelle approximated a second-order problem on the torus using suitable fourth-order equations to conclude. However, using a fourth-order approach imposes several limitations. For instance, initial values have to be sufficiently regular and the treatment of systems is very limited (at best). Also, it seems difficult to work in other domains than the torus. Our novel ingredient to overcome these limitations are a-priori Lp-estimates in omega and t for variational stochastic elliptic problems. They are based on a perturbation argument using Stein interpolation first employed in a work of Böhnlein and Egert. The talk is based on joint work with Mark Veraar and will be accessible also for a deterministic audience with less experience in SPDEs.