The Navier-Stokes equations: on the existence of a weak solution enjoying the energy equality

TL;DR

This paper proves the existence of a weak solution to the Navier-Stokes equations that satisfies the energy equality almost everywhere in time, under certain initial conditions, refining previous results in the field.

Contribution

It establishes the existence of weak solutions with energy equality for the Navier-Stokes equations under minimal initial data assumptions, improving upon prior results.

Findings

Existence of weak solutions satisfying energy equality almost everywhere.

Refinement of previous results on Navier-Stokes solutions.

Solution properties depend on initial data norm.

Abstract

Under the assumption of an initial datum divergence free and in L2, we prove the existence of a weak solution to the Navier-Stokes initial boundary value problem enjoying the energy equality on (0,t), almost everywhere in t>0, in particular, for all t $\in [\theta, \infty)$, with $\theta := \theta(||v0||2)$. Also, the result allows us to refine some others.