A note on the pressure of strong solutions to the Stokes system in bounded and exterior domains

Abstract

We consider the Stokes problem in an exterior domain $\Omega \subset {\mathbb{R}}^n$ with an external force $\bbf \in L^s(0,T; \bW^{k,\, r}(Ω))\, (k\in \N, 1<r<\infty)$. In the present paper we show that in contrast to $\bu$ the boundary regularity of the pressure can be improved according to the differentiability of $\bbf$ up to order $k$. In particular, this implies that the pressure is smooth with respect to $x\in \Omega$ if $\bbf$ is smooth with respect to $x\in \Omega$.