Regularity theory for nonlinear systems of SPDEs

Abstract

We consider systems of stochastic evolutionary equations of the type

$$d{\bf u} = {\rm div}\,{\bf S}(\nabla {\bf u})\,dt + \Phi({\bf u})d{\bf W}_t$$

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

$${\bf S}(\mathbf{\xi}) = (1 + |\mathbf{\xi}|)^{p-2} \mathbf{\xi}, \quad \mathbf{\xi} \in \mathbb{R}^{d \times D},$$

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:

$$\mathbb{E}\bigg[\mathop{\sup}\limits_{t \in (0, T)} \int_{G'}|\nabla{\bf u}(t)|^2\,dx + \int_0^T \int_{G'}|\nabla{\bf F}(\nabla{\bf u})|^2\,dx\,dt\bigg] < \infty,$$

where ${{\bf F}(\mathbf{\xi}) = (1 + |\mathbf{\xi}|)^{\frac{p-2}{2}} \mathbf{\xi}}$ . If we have Uhlenbeck-structure then ${\mathbb{E}\big[\|\nabla{\bf u}\|_q^q\big]}$ is finite for all ${q < \infty}$ if the same is true for the initial data.