RemarkRemark: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Weak existence and smooth regularity scope

Remark

Under Countable Choice, the periodic existence theorem gives a global weak solution for every stated initial datum and positive viscosity; it does not prove global smoothness or uniqueness among all three-dimensional weak solutions. Under Countable Choice and Dependent Choice, the weak–smooth theorem yields uniqueness only on an interval where a smooth solution with the same data is already given. The finite-mode compactness lemma assumes Countable Choice, and the Galerkin compactness argument uses a positive-viscosity gradient bound; that proof does not apply at $ν = 0$ . These are limits of the claims proved on this page and are not assertions about empirical fluid behavior.

Remark references

Leonid Ryzhik, Math 256B Lecture Notes, §§5.6 and 6.1, pp. 52–54.

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions, §§3–5, pp. 5–16.

Facts & Assumptions

Given: Only the theorem statements and proof hypotheses on this page are being summarized.

[F1]

Global periodic weak viscous existence: the conclusion is global unforced weak existence for $d \geq 2$ , $ν > 0$ , $u_{0} \in H$ , and contains no uniqueness or smoothness assertion.

[F2]

Fourier compactness with coefficient time control: under Countable Choice, the compactness statement uses an integrated gradient bound, which in the Galerkin energy estimate is supplied by the positive factor $ν$ .

[F3]

Weak–smooth uniqueness on a smooth interval: under Countable Choice and Dependent Choice, uniqueness is conditional on a smooth comparison solution on the interval.

Proof

1.1givenF1F3

The first sentences restate the exact scope, assumptions, and quantifiers of [F1] and [F3]; neither theorem's conclusion contains global regularity or unrestricted three-dimensional uniqueness.

2.1givenF2algebra∎

The compactness lemma requires the bound $B = \sup_{n} \int_{0}^{T} ∥ \nabla z_{n} ∥_{2}^{2} d t$ ; in the viscous Galerkin energy identity this is obtained after dividing by $2 ν > 0$ . At $ν = 0$ that step supplies no gradient control, so the stated compactness proof gives no inviscid conclusion. This is a proof-scope observation only.

Weak existence and smooth regularity scope

Remark

Remark references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources