A Unified Theory for Some Non-Newtonian Fluids Under Singular Forcing

Abstract

We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain nondegenerate power-law or Carreau-type fluids. Within our setting, 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 [M. Bulíček, L. Diening, and S. Schwarzacher, Anal. PDE, 9 (2016), pp. 1115--1151] 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.