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

==== To legitimately conclude “no finite-time blowup” for a classical solution you need one of these: ====
# Prove strong convergence of approximants in energy-critical norms. Show your Picard/Galerkin sequence converges in norms sufficient to pass the energy equality and higher Sobolev norms. That requires uniform a-priori bounds in those norms or compactness arguments that give strong limits (hard).
# Show the limit is smooth by independent estimates, not by appealing to extended LLL. For example, obtain a uniform bound on a critical norm that rules out blowup (this is the actual content of the Clay problem). Standard conditional results are of the form: if ∫0T∗∥ω(⋅,t)∥L∞ dt<∞\int_0^{T^*}\|\omega(\cdot,t)\|_{L^\infty}\,dt<\infty∫0T∗∥ω(⋅,t)∥L∞dt<∞ (Beale–Kato–Majda) or Serrin-type integrability holds, then no blowup occurs. But you must verify such condition.
# If you use an extended algebra / limit operator LLL then you must explicitly prove LLL commutes with the relevant nonlinearities/integrals and preserves classicality (strong continuity-type axioms). Only then can you pass the energy equality through LLL. This moves the difficulty to foundational axioms; it is not a standard accepted resolution.