Anomalous dissipation for the forced 3D Navier-Stokes equations

TL;DR

This paper demonstrates that for the forced 3D Navier-Stokes equations, there exist classical solutions with energy dissipation rates that do not vanish as viscosity approaches zero, highlighting anomalous dissipation phenomena.

Contribution

The study provides the first explicit construction of solutions exhibiting persistent energy dissipation independent of viscosity in the forced 3D Navier-Stokes equations.

Findings

Existence of solutions with non-vanishing dissipation as viscosity tends to zero

Dissipation rate remains bounded away from zero despite bounded body forces

Results contribute to understanding turbulence and anomalous dissipation in fluid dynamics

Abstract

In this paper, we consider the forced incompressible Navier-Stokes equations with vanishing viscosity on the three-dimensional torus. We show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero, uniformly in the viscosity parameter, while the body forces are uniformly bounded in some reasonable regularity class.