2015): Regularity theory for nonlinear systems of SPDEs
Dominic Breit · 2016
We consider systems of stochastic evolutionary equations of the type $$d{\bf u} = {\rm div}\,{\bf S}( abla {\bf u})\,dt + \Phi({\bf u})d{\bf W}_t$$ 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: $$\mathbb{E}\bigg[\mathop{\sup}\limits_{t \in (0, T)} \int_{G'}| abla{\bf u}(t)|^2\,dx + \int_0^T \int_{G'}| abla{\bf F}( abla{\bf u})|^2\,dx\,dt\bigg] < \infty,$$ where $${{\bf F}(\mathbf{\xi}) = (1 + |\mathbf{\xi}|)^{\frac{p-2}{2}} \mathbf{\xi}}$$ . If we have Uhlenbeck-structure then $${\mathbb{E}\big[\| abla{\bf u}\|_q^q\big]}$$ is finite for all $${q < \infty}$$ if the same is true for the initial data.