A unified theory for some non Newtonian fluids under singular forcing

TL;DR

This paper develops a comprehensive mathematical framework for steady, incompressible non-Newtonian fluids under singular external forces, establishing existence, regularity, and uniqueness results in weighted Lebesgue spaces.

Contribution

It generalizes previous models by providing a full-range theory including weighted space estimates and introduces new technical lemmas for solenoidal, weighted analysis.

Findings

Proved existence and uniqueness of solutions in weighted Lebesgue spaces.

Extended the theory to forces with regularity below the duality exponent.

Introduced solenoidal, weighted, biting div-curl lemma and Lipschitz approximation techniques.

Abstract

We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain non-degenerate power-law or Carreau-type fluids. We provide the full-range theory, namely existence, optimal regularity and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely the weighted Lebesgue spaces, with weights in a respective Muckenhoupt class. The analytical highlight is derivation of existence and uniqueness theory for forcing with its regularity well-below the natural duality exponent, via estimates in weighted spaces. It is a generalization of [Bul\'i\v{c}ek, Diening, Schwarzacher] to incompressible fluids. Moreover, two technical results, needed for our analysis, may be useful for further studies. They are: the solenoidal, weighted, biting div-curl lemma and the solenoidal Lipschitz approximations on domains.