On the energy equality for the evolution Navier–Stokes equations

Integral sufficient conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nabla u"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\nabla u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and mixed, to guarantee the energy equality, EE, for solutions of the Navier–Stokes equations are discussed. An innovative interpretation of some main parameters is revisited, which makes it possible to overcome some apparent incongruence, and also present a simpler, self-contained, proof of the main related result. Furthermore, the above results are used to exhibit in the last section below a typical condition on the direction of vorticity that implies the energy equality.