Global estimates of Generalized Non-Newtonian Stokes systems on non-smooth domains

TL;DR

This paper establishes global gradient and pressure estimates for a generalized non-Newtonian Stokes system with Orlicz growth on non-smooth domains, extending Calderon-Zygmund theory to complex boundary conditions.

Contribution

It introduces a Calderon-Zygmund type estimate for a nonstandard Stokes system with Orlicz growth on irregular domains, addressing boundary flatness and non-Lipschitz regularity.

Findings

Derived global gradient estimates under small BMO non-linearity.

Achieved pressure estimates in non-smooth domains.

Extended Calderon-Zygmund theory to non-Newtonian fluids with complex boundary conditions.

Abstract

We study a generalized Stokes system with Orlicz growth which is nonstandard in a non-smooth domain. Our purpose is to derive a Calderon-Zygmund type estimate of the gradient of a solution and the pressure to such a system like (1.1) under a small BMO non-linearity and a sufficient flatness on the boundary of the domain. In the process, we overcome not only lack of Lipschitz regularity for the corresponding limiting problem, but also the fact that the associated structure depends only on the symmetric part of the gradient for the desired global estimate, which is new even in the literature of elliptic system on a nonsmooth domain.