Anomalous Dissipation for the d-dimensional Navier-Stokes Equations

TL;DR

This paper investigates the vanishing viscosity limit of the d-dimensional Navier-Stokes equations, providing rigorous examples where solutions exhibit anomalous dissipation with energy dissipation remaining positive.

Contribution

It offers the first simple rigorous examples demonstrating anomalous dissipation in solutions of the Navier-Stokes equations as viscosity vanishes.

Findings

Existence of solutions with positive dissipation rate

Demonstration of anomalous dissipation in the vanishing viscosity limit

Rigorous examples of initial data leading to energy dissipation

Abstract

The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier--Stokes equations in the whole space: \begin{equation*} \begin{cases} \partial_tu^\varepsilon+u^\varepsilon\cdot \nabla u^\varepsilon-\varepsilon\Delta u^\varepsilon+\nabla p^\varepsilon=0,\\ \mathrm{div}\ u^\varepsilon=0. \end{cases} \end{equation*} We aim to presenting a simple rigorous examples of initial data which generates the corresponding solutions of the Navier--Stokes equations do exhibit anomalous dissipation. Precisely speaking, we show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero.