Regularity theory for nonlinear systems of SPDEs

We consider systems of stochastic evolutionary equations of the type

du = div S(del u) dt + Phi(u)dW(t)

where S is a non-linear operator, for instance the p-Laplacian

S(xi) = (1 + vertical bar xi vertical bar)(p-2)xi, xi is an element of R-dxD,

with p is an element of (1, infinity) and Phi grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity:

E[sup(t is an element of(0, T)) integral(G')vertical bar del u(t)vertical bar(2) dx + integral(T)(0) integral(G')vertical bar del F(del u)vertical bar(2) dx dt ] <infinity,

where F(xi) = (1 + vertical bar xi vertical bar)(p-2/2) xi. If we have Uhlenbeck-structure then E[parallel to del u parallel to(q)(q)] is finite for all q <infinity if the same is true for the initial data.