Regularity theory for nonlinear systems of SPDEs

We consider systems of stochastic evolutionary equations of the type $du=\mathrm{div}\,S(\nabla u)\,dt+Φ(u)dW_t$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $S(ξ)=(1+|ξ|)^{p-2}ξ,\quad ξ\in\mathbb R^{d\times D},$ with $p\in(1,\infty)$ and $Φ$ grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity: $\mathbb E\bigg[\sup_{t\in(0,T)}\int_{G'}|\nabla u(t)|^2\,dx+\int_0^T\int_{G'}|\nabla F(\nabla u)|^2\,dx\,dt\bigg]<\infty,$ where $F(ξ)=(1+|ξ|)^{\frac{p-2}{2}}ξ$. If we have Uhlenbeck-structure then $\mathbb E\big[\|\nabla u\|_q^q\big]$ is finite for all $q<\infty$.

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