Temporal Regularity of Symmetric Stochastic p-Stokes Systems

We study the symmetric stochastic p-Stokes system, \(p \in (1,\infty )\) $p\in (1,\infty )$ , in a bounded domain. The results are two-fold: First, we show that in the context of analytically weak solutions, the stochastic pressure—related to non-divergence free stochastic forces—enjoys almost \(-1/2\) $-1/2$ temporal derivatives on a Besov scale. Second, we verify that the velocity u of strong solutions obeys 1/2 temporal derivatives in an exponential Nikolskii space. Moreover, we prove that the non-linear symmetric gradient \(V(\mathbb {\epsilon } u) = (\kappa + \left| \mathbb {\epsilon } u\right| )^{(p-2)/2} \mathbb {\epsilon } u\) $V(ϵu)={(\kappa +\left|ϵ,u\right|)}^{(p-2)/2}ϵu$ ,  \(\kappa \ge 0\) $\kappa \ge 0$ , which measures the ellipticity of the p-Stokes system, has 1/2 temporal derivatives in a Nikolskii space.