Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

==== In practice, existence proofs use approximations unu^nun (Galerkin, spectral truncations, or the Picard iteration finite-stage terms). For each approximant (which is smooth, finite-dimensional) the energy equality holds exactly. To pass the equality to a limit uuu you must justify the limit passage carefully. ====

Two standard outcomes:
# Strong convergence case (nice): If un→uu^n\to uun→u strongly in L∞(0,T;L2)L^\infty(0,T;L^2)L∞(0,T;L2) and ∇un→∇u\nabla u^n\to \nabla u∇un→∇u strongly in L2(0,T;L2)L^2(0,T;L^2)L2(0,T;L2), then you can pass equality to the limit and obtain the same energy equality for uuu.
# Weak convergence case (Leray weak solution): Usually you can only extract un⇀uu^n\rightharpoonup uun⇀u weakly in L2(0,T;H1)L^2(0,T;H^1)L2(0,T;H1) and weak-* in L∞(0,T;L2)L^\infty(0,T;L^2)L∞(0,T;L2). In that setting you can pass to the limit in the viscous term only as a lower semicontinuity inequality, and you get the energy inequality 12∥u(t)∥L22+ν∫0t∥∇u(s)∥L22 ds≤12∥u0∥L22,\frac12\|u(t)\|_{L^2}^2 + \nu\int_0^t \|\nabla u(s)\|_{L^2}^2\,ds \le \frac12\|u_0\|_{L^2}^2,21∥u(t)∥L22+ν∫0t∥∇u(s)∥L22ds≤21∥u0∥L22, instead of an equality. That is the standard Leray weak-solution conclusion. You do not obtain equality unless you have additional strong convergence or additional regularity.

So to obtain equality in the limit you must provide strong convergence control on the approximants or additional regularity of the limit.