Temporal regularity of symmetric stochastic $p$-Stokes systems

TL;DR

This paper investigates the temporal regularity of solutions to the symmetric stochastic p-Stokes system, revealing fractional derivatives for pressure and velocity components in specific function spaces, advancing understanding of stochastic fluid dynamics.

Contribution

It establishes fractional temporal regularity results for pressure and velocity in stochastic p-Stokes systems, including weak and strong solutions, using Besov and Nikolskii space frameworks.

Findings

Stochastic pressure has almost -1/2 order temporal derivatives in Besov spaces.

Velocity component u exhibits 1/2 order temporal derivatives in exponential Nikolskii spaces.

Non-linear symmetric gradient V(εu) has 1/2 order temporal derivatives in Nikolskii space.

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 $V(\varepsilon u) = (\kappa + |\varepsilon u|)^{(p-2)/2} \varepsilon u$, $\kappa \geq 0$, has $1/2$ temporal derivatives in a Nikolskii space.