# On the Kolmogorov dissipation law in a damped Navier-Stokes equation

## TL;DR

This paper investigates the impact of a damping term on the energy dissipation and turbulence in Navier-Stokes equations, showing that damping can suppress turbulence despite high Reynolds numbers, aligning with Kolmogorov's theory.

## Contribution

It introduces a damping-modified Navier-Stokes model, analyzes its well-posedness, and reveals conditions where damping suppresses turbulence even at high Reynolds numbers.

## Key findings

- Energy dissipation rate bounds consistent with Kolmogorov K41 theory.
- Damping can eliminate turbulence despite large Reynolds numbers.
- Weak solutions exhibit behavior similar to classical turbulence theory.

## Abstract

We consider here the Navier-Stokes equations in $\mathbb{R}^{3}$ with a stationary, divergence-free external force and with an additional damping term that depends on two parameters. We first study the well-posedness of weak solutions for these equations and then, for a particular set of the damping parameters, we will obtain an upper and lower control for the energy dissipation rate $\varepsilon$ according to the Kolmogorov K41 theory. However, although the behavior of weak solutions corresponds to the K41 theory, we will show that in some specific cases the damping term introduced in the Navier-Stokes equations could annihilate the turbulence even though the Grashof number (which are equivalent to the Reynolds number) are large.