On the existence of weak solutions for steady flows of generalized viscous fluids

TL;DR

This paper proves the existence of weak solutions for steady flows of generalized viscous fluids modeled by variable exponent Navier-Stokes equations, extending previous results by allowing the lowest possible exponent bound without additional regularity assumptions.

Contribution

It establishes the existence of weak solutions for variable exponent flows with minimal exponent bounds, improving prior results by removing regularity constraints on the exponent function.

Findings

Existence of weak solutions for variable exponent flows with q ≥ α > 2N/(N+2)

No additional regularity assumptions on the exponent q are needed

The minimal possible bound for q is attained in the existence proof

Abstract

In this work we investigate the existence of weak solutions for steady flows of generalized incompressible and homogeneous viscous fluids. 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 prove the existence of weak solutions for any variable exponent $q\geq\alpha>\frac{2N}{N+2}$, where $\alpha=\mathrm{ess}\inf q$. This work improves all the known existence results in the sense that the lowest possible bound of $q$ is attained and no other assumption on the regularity of $q$ is required.