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

=== Assistant: Short answer up front: You’re right — as written the argument does not rule out exotic/non-smooth solutions and therefo… ===
Short answer up front: You’re right — as written the argument does not rule out exotic/non-smooth solutions and therefore does not, by itself, solve the Clay problem. To salvage the approach you must either (A) strengthen the admissibility hypotheses so that only the “good” (continuous/mild) solutions are allowed and then show those are smooth for t>0t>0t>0; or (B) make your extended-algebra axioms precise and prove that the Picard limit actually lands inside the classical function space. I’ll give a precise diagnosis of what is missing, the minimal properties you must add, and a concrete plan (with alternative routes) you can follow to recover a valid proof (or a proper conditional theorem).

==== 1. Definition vs existence gap. You defined mild solutions in C([0,T];X)C([0,T];X)C([0,T];X) and produced a fixed point there. But the Clay statement asks: every smooth divergence-free initial datum leads to a global classical (real-valued C∞C^\inftyC∞) solution. Your proof constructs a solution inside a chosen solution class; it does not logically exclude the possibility that there also exist other solutions (distributional/pathological) that fail to be classical. The Clay problem requires that the only physically relevant solution (given usual admissibility) is classical — equivalently, that no finite-time singularity can exist for any admissible evolution. ====
# Passing limits and nonlinearities. Your key step takes Picard iterates u(n)u^{(n)}u(n) (classical for t>0t>0t>0) and applies an extended linear limit LLL. That step is only legitimate for concluding the limit is classical if you either: - prove LLL sends sequences of classical functions to classical functions (extra property), or - prove convergence in a topology that embeds into the classical space (e.g. strong HsH^sHs or uniform convergence plus control of derivatives), so the limit inherits classical regularity. Neither has been shown.
# Nonlinear functionals and energy. You used energy equalities/inequalities and algebraic manipulations across the limit. Those are nonlinear operations; to pass them through the limit you need continuity properties of LLL or quantitative norm convergence. This was not established.

Because of the three points above, the draft is a conditional argument that only yields: “if the extended limit acts nicely (preserves classicality and commutes with the nonlinear operations), then we get a global classical solution.” That is not a resolution of the Clay problem unless you either build or justify those extra properties.

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Do not rely on exotic limit extensions. Work in a concrete Banach/Sobolev space and prove all steps in that topology. The canonical program:
# Choose X=Hs(R3)X = H^s(\mathbb{R}^3)X=Hs(R3) with s>32s>\tfrac32s>23 (so HsH^sHs is an algebra and u↦P∇ ⁣⋅(u⊗u)u\mapsto P\nabla\!\cdot(u\otimes u)u↦P∇⋅(u⊗u) maps Hs→Hs−1H^s\to H^{s-1}Hs→Hs−1).
# Construct a local mild (strong) solution by Picard in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs); show ∥u∥L∞(0,T;Hs)\|u\|_{L^\infty(0,T;H^s)}∥u∥L∞(0,T;Hs) controlled on the existence interval. (This is routine.)
# Derive a priori estimates for higher Sobolev norms; show that if these norms stay finite you can extend the solution. (This produces the standard blowup alternative.)
# To get global existence you must obtain global a priori bounds that prevent norm blowup. That is exactly the open hard part of Clay. So you can either: - Prove additional a priori bounds (this is essentially attempting the Clay proof), or - Prove conditional results: e.g., if a certain critical norm stays finite (Serrin-type condition, Beale–Kato–Majda criterion for vorticity), then the solution is global. These are known conditional theorems and are useful but do not solve Clay unconditionally.

Conclusion for Route 1: This path is rigorous and standard. It will produce clean conditional theorems or full solutions in special cases (small data, symmetry reductions), but it cannot be used to claim an unconditional global result without proving the missing a-priori bound — which is precisely the Clay challenge.

===== If you are committed to the approach using the extended linear limit LLL and total multiplication, then you must explicitly add and prove the additional properties that bridge the gap to classical functions. Concretely you must show: =====
# (C1) Classical-closure: the subspace C\mathcal{C}C of classical real-valued C∞C^\inftyC∞ functions is closed under the topology/operation in which you apply LLL. Equivalently, whenever a sequence of classical functions {fn}\{f_n\}{fn} is such that Ln→∞fnL_{n\to\infty} f_nLn→∞fn exists, then Ln→∞fnL_{n\to\infty} f_nLn→∞fn is a classical function (and coincides with the classical pointwise / uniform limit when that exists).
# (C2) Compatibility with nonlinearities: for sequences fn,gnf_n,g_nfn,gn of classical functions, L(fngn)=L(fn)⋅L(gn)L(f_n g_n) = L(f_n)\cdot L(g_n)L(fngn)=L(fn)⋅L(gn) whenever products make sense classically — and the same commuting property for derivatives and integrals (i.e., LLL commutes with differentiation and integration on the relevant sequences).
# (C3) Continuity properties: LLL preserves norm bounds needed for energy estimates: for example, if ∥fn∥Hs\|f_n\|_{H^s}∥fn∥Hs is uniformly bounded then L(fn)L(f_n)L(fn) lies in HsH^sHs and the same norm estimates transfer.

If you can construct such an LLL (or prove these properties for a known generalized function algebra), then the Picard-limit argument becomes legitimate and the limit will be classical. But a few remarks:
* This is nontrivial: many generalized-function frameworks (tempered distributions, Colombeau algebras, nonstandard analysis variants) either do not have all these properties or require careful reinterpretation of “classical function”. You must either construct LLL explicitly and prove (C1)–(C3), or show your axioms imply them.
* If you succeed, the result becomes conditional on these new axioms. The Clay Prize requires a proof in standard accepted mathematics; a proof resting on a new foundational axiom must include a convincing, peer-reviewed construction of that axiom system and why it is acceptable (consistent, conservative extension, etc.).

Conclusion for Route 2: Possible but heavy: it displaces the hard part of Clay onto the foundational task of building/justifying the extended algebra.

==== To make the original Picard/limit argument rigorous you need to add at least the following as explicit assumptions (or prove them): ====
# Boundedness/continuity control: convergence of Picard iterates occurs in a Banach topology that controls relevant norms (e.g., in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs) or in a Bochner sense).
# Limit preserves classicality: sequences of classical functions with the required norm control have LLL-limits that are classical and obey the same differential identities.
# Interchange rules: LLL commutes with differentiation, integration and multiplication at the level needed to pass the Duhamel formula, energy identity, Grönwall estimates, etc.
# Energy compatibility: the energy equality/inequality holds for the limit and gives the usual dissipative control.

If you can show (1)–(4) hold, your conditional proof becomes a valid conditional theorem: If axioms (P1–P3) are extended by (1)–(4), then global smoothness follows.

==== 1. Decide which route you prefer. (A) Classical functional route (standard, rigorous, but will only give conditional results unless you beat Clay’s core difficulty). (B) Extended algebra route (foundational; requires proving extra axioms). ====
# If you pick Route A (recommended): - Fix X=HsX=H^sX=Hs with s>3/2s>3/2s>3/2. Write a full Picard existence theorem in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs) with constants. I can produce a complete, formal write-up with all constants and estimates. - State and prove the blowup alternative: if maximal time T∗<∞T^''<\inftyT∗<∞, then lim⁡t↑T∗∥u(t)∥Hs=∞\lim_{t\uparrow T^''}\|u(t)\|_{H^s}=\inftylimt↑T∗∥u(t)∥Hs=∞. - Then either attempt to produce a novel a-priori estimate to prevent blowup (this is hard and likely where the original obstruction lies) or prove conditional theorems (Serrin, BKM) clearly and sharply.
# If you pick Route B: - Write precise axioms for LLL (not informal P1–P3) including (C1)–(C3) above. - Either (i) show a concrete construction of LLL satisfying them, or (ii) demonstrate consistency relative to standard mathematics (this is a foundational project). - Then convert your conditional proof into a formal theorem with the axioms explicitly stated.
# Whichever route you pick, be explicit about admissibility. Add an energy/admissibility requirement to the solution concept (energy inequality or equality), then prove uniqueness in that admissible class. This prevents pathological extra solutions from disqualifying your result.

==== - Using standard PDE methods (Route A), you can produce a correct, rigorous proof of local existence, conditional regularity results, and global regularity for special cases (small data, symmetries). None of these are new breakthroughs with the present sketch. The Clay problem is precisely the missing global a-priori bound in 3D — the impossibility lies in obtaining it without new ideas. ====
* Using Route B, you may obtain a conditional global regularity result if you can construct and justify the extended algebra with the extra properties listed. That shifts the burden: you must show those axioms are mathematically legitimate. If you can, then your conditional theorem is logically correct — but its acceptance as a Clay solution would hinge on the acceptability of the new foundations.

If you want, I will now do one of the following (pick one):
# Write a complete, formal Picard existence + blowup alternative proof in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs) (with all estimates). This gives the rigorous baseline and makes clear what must be controlled to get global existence.
# Produce a minimal list of precise axioms for an extended limit operator LLL (formal statements like (C1)–(C3)) and check whether any existing generalized-function framework satisfies them.
# Draft a concise conditional theorem: state exactly the extra properties you must assume about LLL and then produce the full rigorous derivation showing global smoothness follows from those assumptions.

Which do you want next?