On the energy equality for the 3D Navier-Stokes equations

TL;DR

This paper investigates conditions under which solutions to the 3D Navier-Stokes equations conserve energy, introducing new criteria involving velocity gradients and analyzing very-weak solutions within the Shinbrot class.

Contribution

It establishes new energy conservation criteria for Leray-Hopf weak solutions and extends energy equality results to very-weak solutions in the Shinbrot class.

Findings

New criteria involving the velocity gradient for energy conservation.

Comparison with scaling invariant spaces and Onsager conjecture.

Energy equality proven for solutions in the Shinbrot class.

Abstract

In this paper we study the problem of energy conservation for the solutions of the initial boundary value problem associated to the 3D Navier-Stokes equations, with Dirichlet boundary conditions. First, we consider Leray-Hopf weak solutions and we prove some new criteria, involving the gradient of the velocity. Next, we compare them with the existing literature in scaling invariant spaces and with the Onsager conjecture. Then, we consider the problem of energy conservation for very-weak solutions, proving energy equality for distributional solutions belonging to the so-called Shinbrot class. A possible explanation of the role of this classical class of solutions, which is not scaling invariant, is also given.