A high regularity result of solutions to modified p-Stokes equations

TL;DR

This paper establishes high regularity results for solutions to a modified p-Stokes system, a perturbed p-Laplacean system, demonstrating that solutions are highly smooth and continuous, even in unbounded domains.

Contribution

It provides the first high regularity results for solutions of the coupled (u, π) system, including unbounded domains, extending previous regularity theories.

Findings

Solutions have second derivatives in L^q spaces.

Pressure and velocity are Hölder continuous.

Results apply to both modified p-Stokes and p-Laplacean systems.

Abstract

This paper is concerned with a special elliptic system, which can be seen as a perturbed $p$-Laplacean system, $p\in(1,2)$, and, for its "shape", it is close to the $p$-Stokes system. Since our "stress tensor" is given by means of $\nabla u $ and not by its symmetric part, then our system is not a $p$-Stokes system. Hence, the system is called {\it modified} $p$-Stokes system. We look for the high regularity of the solutions $(u,\pi)$, that is $D^2u,\nabla\pi \in L^q,q\in(1,\infty)$. In particular, we get $\nabla u,\pi\in C^{0,\alpha}$. As far as we know, such a result of high regularity is the first concerning the coupling of unknowns $(u,\pi)$. However, our result also holds for the $p$-Laplacean, and it is the first high regularity result in unbounded domains.