  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 $q<\infty$.
