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

=== Assistant: Good — these are the two crucial points that determine whether the whole argument is valid or circular. ===
Good — these are the two crucial points that determine whether the whole argument is valid or circular. I’ll answer them plainly and precisely.

Short answer — what is wrong and how to fix it
# What’s wrong with the earlier “we have smoothness of uuu and ppp already proved above” claim: that step was circular. You did not prove classical smoothness of the limit solution in a standard topology; you only produced a limit in an extended algebra (or asserted termwise differentiation under an extended limit LLL). Without a separate justification that the Picard iterates converge strongly in a Banach/Sobolev space (or that the extended limit preserves membership in the classical function space and commutes with nonlinear operations), you do not have a classical solution. Therefore you cannot legitimately apply the standard energy equality for classical solutions to the limit. In short: you proved global bounds only inside your extended framework or under extra axioms, not for a classical Navier–Stokes solution in the usual sense. That is the key gap.
# Why exchanging ∂t\partial_t∂t and the time integral in the energy derivation is legitimate for smooth solutions but not for a limit without justification: For a genuinely smooth solution the manipulations are standard and fully justified; for a limit constructed via LLL or by weak convergence you must verify the hypotheses of differentiation-under-the-integral (or use approximation + dominated convergence / Bochner theorems). If you lack strong convergence or a continuity property of LLL, you cannot push derivatives or nonlinear operations through the limit.

Below I give the precise, standard derivation of the energy equality for classical smooth solutions (with exact hypotheses needed), then explain how the same step must be justified when you work with approximations and limits (Galerkin / Picard sequences), and finally explain the exact places your earlier draft is insufficient and how to repair them.

==== Hypotheses. Let u(x,t)u(x,t)u(x,t) be a sufficiently smooth solution on [0,T]×R3[0,T]\times\mathbb{R}^3[0,T]×R3 (for example u∈C1([0,T];L2)∩C([0,T];H2)u\in C^1([0,T];L^2)\cap C([0,T];H^2)u∈C1([0,T];L2)∩C([0,T];H2); decays at infinity fast enough or use periodic/compact domain so boundary terms vanish). Let viscosity ν>0\nu>0ν>0. Assume ∇⋅u=0\nabla\cdot u=0∇⋅u=0. ====

Claim (energy equality).

12ddt∥u(t)∥L22+ν∥∇u(t)∥L22=0.\frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^2}^2 + \nu\|\nabla u(t)\|_{L^2}^2 = 0.21dtd∥u(t)∥L22+ν∥∇u(t)∥L22=0.
Proof. Multiply the PDE by uuu and integrate over space:
# Start from the momentum equation

∂tu−νΔu+(u⋅∇)u+∇p=0.\partial_t u - \nu\Delta u + (u\cdot\nabla)u + \nabla p = 0.∂tu−νΔu+(u⋅∇)u+∇p=0.
# Take the L2L^2L2-inner product with uuu (i.e. integrate componentwise: ∫(⋯ )⋅u dx\int (\cdots)\cdot u\,dx∫(⋯)⋅udx). Each term:
* Time-derivative term: ∫(∂tu)⋅u dx=12ddt∫∣u∣2 dx=12ddt∥u∥L22,\int (\partial_t u)\cdot u\,dx = \tfrac12\frac{d}{dt}\int |u|^2\,dx = \tfrac12\frac{d}{dt}\|u\|_{L^2}^2,∫(∂tu)⋅udx=21dtd∫∣u∣2dx=21dtd∥u∥L22, because the map t↦∥u(t)∥L22t\mapsto \|u(t)\|^2_{L^2}t↦∥u(t)∥L22 is differentiable when u∈C1([0,T];L2)u\in C^1([0,T];L^2)u∈C1([0,T];L2) and the derivative equals 2⟨∂tu,u⟩2\langle \partial_t u,u\rangle2⟨∂tu,u⟩.
* Viscous term (integration by parts; boundary terms vanish by decay or periodicity): −ν∫(Δu)⋅u dx=ν∫∣∇u∣2 dx=ν∥∇u∥L22.-\nu\int (\Delta u)\cdot u\,dx = \nu\int |\nabla u|^2\,dx = \nu\|\nabla u\|_{L^2}^2.−ν∫(Δu)⋅udx=ν∫∣∇u∣2dx=ν∥∇u∥L22. (Use ∫(Δu)⋅u=−∫∇u:∇u\int (\Delta u)\cdot u = -\int \nabla u : \nabla u∫(Δu)⋅u=−∫∇u:∇u.)
* Pressure term: ∫(∇p)⋅u dx=−∫p (∇⋅u) dx=0,\int (\nabla p)\cdot u\,dx = -\int p\,(\nabla\cdot u)\,dx = 0,∫(∇p)⋅udx=−∫p(∇⋅u)dx=0, because ∇⋅u=0\nabla\cdot u=0∇⋅u=0.
* Nonlinear convection term: ∫(u⋅∇)u⋅u dx=∑i∑j∫uj∂jui ui dx=12∑j∫uj∂j(∣u∣2) dx=−12∑j∫(∂juj) ∣u∣2 dx=0,\int (u\cdot\nabla)u\cdot u\,dx = \sum_i\sum_j \int u_j \partial_j u_i \, u_i\,dx = \frac12\sum_j\int u_j \partial_j (|u|^2)\,dx = -\frac12\sum_j\int (\partial_j u_j)\,|u|^2\,dx = 0,∫(u⋅∇)u⋅udx=i∑j∑∫uj∂juiuidx=21j∑∫uj∂j(∣u∣2)dx=−21j∑∫(∂juj)∣u∣2dx=0, again because ∇⋅u=0\nabla\cdot u=0∇⋅u=0 (or integrate by parts, boundary terms vanish).
# Summing terms yields exactly

12ddt∥u∥L22+ν∥∇u∥L22=0.\frac12\frac{d}{dt}\|u\|_{L^2}^2 + \nu\|\nabla u\|_{L^2}^2 = 0.21dtd∥u∥L22+ν∥∇u∥L22=0.
This is rigorous whenever the above integrals and integrations by parts are justified — i.e. uuu is smooth enough and decays or the domain has suitable boundary conditions.

==== 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.

==== Your draft applied the extended linear limit LLL or passed to a limit of Picard iterates and then asserted the energy equality holds for the limit. This step is invalid unless you have one of the two things above: ====
* Either you have shown the Picard iterates converge strongly in the norms that control the energy (i.e. strong Lt∞Lx2L^\infty_t L^2_xLt∞Lx2 convergence and strong Lt2Hx1L^2_t H^1_xLt2Hx1 convergence), so you can pass the equality term-by-term;
* Or you have proven that the linear limit operator LLL has the continuity properties required to carry the nonlinear quadratic forms and integrals through — something you have not established.

Without that, you can at best conclude the energy inequality (via weak lower-semicontinuity) not the exact conservation/dissipation equality. And the energy inequality alone is not enough to rule out finite-time blow-up of classical norms (it controls only L2L^2L2 and the dissipation integral).

Hence the claim “no finite-time blowup” is unsupported: you either must prove additional a-priori bounds in supercritical norms or show the limit is classical (strong convergence) so you can upgrade to equality and then bootstrap regularity — neither has been done.

==== 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.

==== - For a genuinely classical smooth solution, the energy equality proof (multiply by uuu, integrate, integrate by parts) and exchanging ∂t\partial_t∂t with the space integral are fully justified under the standard smoothness hypotheses (e.g. u∈Ct1Lx2∩CtHx2u\in C^1_t L^2_x \cap C_t H^2_xu∈Ct1Lx2∩CtHx2). ====
* But you did not prove that your constructed solution is classical in the usual sense without extra nonstandard axioms or without proving strong convergence in the necessary norms. Therefore you cannot legitimately claim “no finite-time blowup” for the classical Navier–Stokes problem. The step in your draft that carries energy equality from approximants to the limit is the hidden, unjustified leap.

==== (standard sources that treat these limit passages and energy arguments rigorously) ====
* Temam, Navier–Stokes Equations: Theory and Numerical Analysis — Galerkin method, energy inequality, weak solutions.
* Constantin & Foias, Navier–Stokes Equations — functional framework and energy methods.
* Pazy, Semigroups of Linear Operators — Bochner integrals and differentiation under the integral in Banach spaces.

If you want, I can now do one of the following concrete things (pick one):
# Write a full, rigorous Galerkin approximation → energy equality for approximants → passage to the limit showing only the energy inequality holds unless you have strong convergence. (This will show exactly where equality drops to an inequality.)
# Show precisely what extra strong-convergence estimates you would need on the Picard iterates to pass the equality to the limit, and give a roadmap for how to try to prove them.
# Formalize the exact extra axioms the extended-limit operator LLL would need so that it legitimately commutes with nonlinearities and integrals (and show why those axioms are strong).

Which of these do you want next?