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

==== - Gap 1 — Passing limits through nonlinearities/integrals. Strong convergence in the necessary norms must be proven; an algebraic extended-limit LLL without continuity does not justify interchanging limit and integral or passing nonlinear quantities to the limit. Consequence: you cannot deduce the energy equality or higher bounds for the limit unless you have strong convergence or continuity properties of the limit operator. ====
* Gap 2 — Energy controls only L2L^2L2. Energy equality gives dissipation but only controls the L2L^2L2-norm and the time integral of ∥∇u∥L22\|\nabla u\|_{L^2}^2∥∇u∥L22. Those are insufficient to bound supercritical norms (like HsH^sHs, s>1s>1s>1), so blowup in HsH^sHs remains possible.
* Gap 3 — Critical norm control is the real obstacle. All successful criteria reduce the problem to proving boundedness of a scaling-critical quantity (e.g. vorticity L∞L^\inftyL∞ integral, Serrin norms, or scaling-invariant Besov norms). Those are nontrivial and are not produced by the above energy bounds.