Existence of weak solutions for the generalized Navier-Stokes equations with damping

TL;DR

This paper proves the existence of weak solutions for a generalized Navier-Stokes model with damping, applicable to non-Newtonian fluids, using advanced mathematical techniques.

Contribution

It establishes the existence of weak solutions for the generalized Navier-Stokes equations with damping for a broad range of exponents, extending previous results.

Findings

Existence of weak solutions for q > 2N/(N+2) and σ > 1.

Application of regularization, monotone operators, and compactness methods.

Results applicable to non-Newtonian fluid flow models.

Abstract

In this work we consider the generalized Navier-Stoke equations with the presence of a damping term in the momentum equation. % The problem studied here derives from the set of equations which govern the isothermal flow of incompressible, homogeneous and non-Newtonian fluids. % For the generalized Navier-Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any $q>\frac{2N}{N+2}$ and any $\sigma>1$, where $q$ is the exponent of the diffusion term and $\sigma$ is the exponent which characterizes the damping term.