On the Energy Equality for Distributional Solutions to Navier-Stokes Equations

TL;DR

This paper demonstrates that the energy equality for distributional solutions to the Navier-Stokes equations holds without requiring the solution to be in the Leray-Hopf class, expanding the class of solutions known to satisfy energy equality.

Contribution

It proves that the Leray-Hopf class assumption is unnecessary for energy equality in solutions belonging to L^4(0,T;L^4).

Findings

Energy equality holds without Leray-Hopf assumption.

Distributional solutions in L^4(0,T;L^4) satisfy energy equality.

The classical requirement for Leray-Hopf solutions is relaxed.

Abstract

A classical result of J.-L. Lions asserts that if a solution to the Navier-Stokes equations is such that: (i) it is in the Leray-Hopf class, and (ii) belongs to $L^4(0,T;L^4)$, then it must satisfy the energy equality in the time interval $[0,T]$. In this note we show that assumption (i) is not necessary.