Temporal Regularity of Symmetric Stochastic p-Stokes Systems

Abstract

AbstractWe study the symmetric stochastic p-Stokes system, $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) , 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 ( ϵ u ) = ( κ + ϵ u ) ( p - 2 ) / 2 ϵ u , $$\kappa \ge 0$$ κ ≥ 0 , which measures the ellipticity of the p-Stokes system, has 1/2 temporal derivatives in a Nikolskii space.