Regularity theory for nonlinear systems of SPDEs

We consider systems of stochastic evolutionary equations of the type $$du=\mathrm{div}\,S(\nabla u)\,dt+\Phi(u)dW_t$$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $$S(\xi)=(1+|\xi|)^{p-2}\xi,\quad \xi\in\mathbb R^{d\times D},$$ with $p\in(1,\infty)$ and $\Phi$ 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 $F(\xi)=(1+|\xi|)^{\frac{p-2}{2}}\xi$. If we have Uhlenbeck-structure then $\mathbb E\big[\|\nabla u\|_q^q\big]$ is finite for all $q<\infty$.