Regularity theory for nonlinear systems of SPDEs

Abstract

We consider systems of stochastic evolutionary equations of the type $$du=\mathrm{div}S(\nabla u)dt+\Phi (u)d{W}_t$$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $$S(\xi )=(1+|\xi |{)}^{p-2}\xi ,\xi \in {\mathbb{R}}^{d\times D},$$ 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: $$\mathbb{E}[\underset{t\in (0,T)}{\sup }{\int }_{G^{\prime }}|\nabla u(t){|}^{2}dx+{\int }_0^T{\int }_{G^{\prime }}|\nabla F(\nabla u){|}^{2}dxdt]<\infty ,$$ where $F(\xi )=(1+|\xi |{)}^{\frac{p-2}{2}}\xi$. If we have Uhlenbeck-structure then $\mathbb{E}[\Vert \nabla u{\Vert }_q^q]$ is finite for all $q<\infty$.

References in corpus (2)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• Solenoidal Lipschitz truncation for parabolic PDE's

Cited by in corpus (2)

• An averaged space-time discretization of the stochastic $p$-Laplace system
• $L^p$-estimates and regularity for SPDEs with monotone semilinearity