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

TL;DR

This paper investigates the regularity of the pressure in the Stokes system within exterior domains, showing that the pressure's smoothness correlates with the differentiability of the external force, unlike the velocity field.

Contribution

It demonstrates that the boundary regularity of the pressure in the Stokes system can be enhanced based on the differentiability of the external force, providing new insights into pressure regularity.

Findings

Pressure regularity improves with force smoothness.

Pressure becomes smooth if the external force is smooth.

Velocity regularity does not necessarily improve with force smoothness.

Abstract

We consider the Stokes problem in an exterior domain $\Omega \subset \R^n$ with an external force $\bbf \in L^s(0,T; \bW^{k,\, r}(\Omega ))\, (k\in \N, 1<r<\infty)$. 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 $.