A note about existence for a class of viscous fluid problems

TL;DR

This paper proves the existence of weak solutions for a class of non-Newtonian fluid problems modeled by generalized Navier-Stokes equations with variable exponent, extending previous results by relaxing regularity conditions on the exponent.

Contribution

It demonstrates that the existence of weak solutions holds for variable exponents without requiring log-Hölder continuity, broadening the applicability of the theory.

Findings

Existence of weak solutions for variable exponent flows.

Relaxation of regularity conditions on the exponent q.

Results valid for q ≥ α > 2N/(N+2).

Abstract

In this work the existence of weak solutions for a class of non-Newtonian viscous fluid problems is analyzed. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the exponent $q$ that characterizes the flow depends on the space variable: $q=q(\mathbf{x})$. For the associated boundary-value problem we show that, in some situations, the log-H\"older continuity condition on $q$ can be dropped and the result of the existence of weak solutions still remain valid for any variable exponent $q\geq\alpha>\frac{2N}{N+2}$, where $\alpha=\mathrm{ess}\inf q$.