Temporal regularity of symmetric stochastic $p$-Stokes systems

Abstract

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

arXiv comment28 pages