Higher differentiability for solutions to stationary $p$-Stokes systems under sub-quadratic growth conditions

TL;DR

This paper proves fractional higher differentiability of solutions to stationary p-Stokes systems with sub-quadratic growth, advancing understanding of regularity in nonlinear fluid mechanics models.

Contribution

It establishes fractional higher differentiability results for both the symmetric gradient and pressure in stationary p-Stokes systems under sub-quadratic growth conditions.

Findings

Proves fractional higher differentiability of the symmetric gradient.

Establishes fractional higher differentiability of the pressure.

Advances regularity theory for nonlinear fluid systems.

Abstract

We consider weak solutions $(u,\pi):\mathbb{R}^n\supset\Omega\to\ \mathbb{R}^n\times\ \mathbb{R}$ to stationary $p$-Stokes systems of the type \[ \begin{cases} -\mathrm{div} (a(\mathcal{E} u))+\nabla\pi=f \\ \mathrm{div}(u)=0, \end{cases} \] in $\Omega,$ where the function $a(\xi)$ satisfies $p$-growth conditions in $\xi$. By $\mathcal{E} u$ we denote the symmetric part of the gradient $Du$. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the gradient $ D u$ and of the pressure $\pi$.