Existence of steady very weak solutions to Navier-Stokes equations with non-Newtonian stress tensors

TL;DR

This paper proves the existence of very weak solutions for steady non-Newtonian fluid flows with non-standard stress tensors, introducing new a-priori estimates and a solenoidal Lipschitz truncation method.

Contribution

It establishes the existence of solutions under broader conditions and develops novel estimates and truncation techniques for non-Newtonian steady flows.

Findings

Existence of very weak solutions for non-Newtonian steady flows.

New a-priori estimates using solenoidal Lipschitz truncation.

Estimates valid even with the convective term present.

Abstract

We provide existence of very weak solutions and new a-priori estimates for steady flows of non-Newtonian fluids when the right-hand sides are not in the natural existence class. To obtain the a-priori estimates we make use of a newly developed solenoidal Lipschitz truncation that preserves zero boundary values. We provide also estimates in (Muckenhoupt) weighted spaces which permit us to regain a duality pairing. Our estimates are valid even in the presence of the convective term. They are obtained via a newly developed comparison method that allows to "cut out" the singularities of the right hand side such that the skew symmetry of the convective term can be used for large parts of the right hand side.