Global continuity and BMO estimates for non-Newtonian fluids with perfect slip boundary conditions

TL;DR

This paper establishes stability and regularity results for the generalized stationary Stokes system with perfect slip boundary conditions, covering a wide class of non-Newtonian fluids including shear thickening and thinning types, with new global estimates.

Contribution

It provides the first global continuity and BMO estimates for solutions of non-Newtonian fluids with slip boundary conditions, extending classical results to more general Orlicz growth conditions.

Findings

Gradients of solutions are globally continuous under certain conditions.

The results include shear thickening and shear thinning fluids, such as power law fluids.

New global estimates are established even for the linear Stokes system.

Abstract

We study the generalized stationary Stokes system in a bounded domain in the plane equipped with perfect slip boundary conditions. We show natural stability results in oscillatory spaces, i.e. H\"older spaces and Campanato spaces including the border-line spaces of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In particular, we show that under appropriate assumptions gradients of solutions are globally continuous. Since the stress tensor is assumed to be governed by a general Orlicz function, our theory includes various cases of (possibly degenerate) shear thickening and shear thinning fluids; including the model case of power law fluids. The global estimates seem to be new even in case of the linear Stokes system.