Definition
Kinetic-energy inequality
An integrated upper bound allowing energy loss beyond the explicitly recorded viscous dissipation.
A velocity-force pair satisfies the kinetic-energy inequality from time if, for the stated later times ,
The dissipation and forcing integrals must be defined. The domain, time representative, and admissible starting and ending times form part of the condition.
Inequality versus equality
A smooth solution with justified energy integration satisfies equality, hence also this inequality. In a weak-solution definition the inequality is an additional admissibility condition. A limit can lose equality because norm convergence is weaker than strong convergence. Requiring the inequality from the initial time only is different from requiring it from almost every intermediate time as well.
Solution classes
The Leray–Hopf definition states a particular global time convention. The local energy inequality uses nonnegative space-time tests and is a separate condition.