Abstract
We consider systems of stochastic evolutionary equations of the type
where S is a non-linear operator, for instance the p-Laplacian
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:
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.
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Breit, D. Regularity theory for nonlinear systems of SPDEs. manuscripta math. 146, 329–349 (2015). https://doi.org/10.1007/s00229-014-0704-8
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DOI: https://doi.org/10.1007/s00229-014-0704-8