TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 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.

Skew reduced convection and global finite dynamics

Statement

Assume Countable Choice, meaning every sequence of nonempty sets admits a choice function. Let $d \in {1, 2, 3}$ , $r \in N$ with $r \geq 1$ , and $Q_{d} = R^{d} / (2 π Z)^{d}$ with probability measure $d μ = (2 π)^{- d} d x$ . In the supplied Fourier-Sobolev convention, $H^{0} (Q_{d}; R^{d})$ is the completion of finite real trigonometric polynomials in the norm $∥ q ∥_{H^{0}}^{2} = \sum_{k \in Z^{d}} ∣ \hat{q} (k) ∣^{2}$ , with unique $L^{2}$ representatives and conjugate-symmetric coefficients for real classes. Let $v_{1}, \dots, v_{r} \in H^{0} (Q_{d}; R^{d})$ have smooth $2 π$ -periodic representatives satisfying $div v_{i} = 0$ and $\int_{Q_{d}} v_{i} \cdot v_{j} d μ = δ_{i j}$ . Fix $ν \geq 0$ and $a^{0} \in R^{r}$ , set $u_{a} = \sum_{i} a_{i} v_{i}$ , and define

$A_{i j} = \int_{Q_{d}} \nabla v_{j} : \nabla v_{i} d μ, c_{i j k} = \int_{Q_{d}} ((v_{j} \cdot \nabla) v_{k}) \cdot v_{i} d μ,$

where $\nabla v_{j} : \nabla v_{i} = \sum_{ℓ, m = 1}^{d} (\partial_{x_{m}} v_{j}^{ℓ}) (\partial_{x_{m}} v_{i}^{ℓ})$ and $((v_{j} \cdot \nabla) v_{k})^{ℓ} = \sum_{m = 1}^{d} v_{j}^{m} \partial_{x_{m}} v_{k}^{ℓ}$ . Consider the coefficient initial-value system

$a_{i}^{'} (t) + \sum_{j, k = 1}^{r} c_{i j k} a_{j} (t) a_{k} (t) + ν \sum_{j = 1}^{r} A_{i j} a_{j} (t) = 0, a_{i} (0) = a_{i}^{0}, i = 1, \dots, r .$

There is exactly one solution $a \in C^{1} ([0, \infty), R^{r})$ of this system. For every $t \geq 0$ , it satisfies

$\frac{1}{2} \frac{d}{d t} ∣ a (t) ∣^{2} + ν \int_{Q_{d}} ∣ \nabla u_{a (t)} ∣^{2} d μ = 0,$

$∣ a (t) ∣^{2} + 2 ν \int_{0}^{t} \int_{Q_{d}} ∣ \nabla u_{a (s)} ∣^{2} d μ d s = ∣ a^{0} ∣^{2} .$

This is a global result for the specified finite coefficient ODE only; it asserts no convergence to, existence of, or uniqueness for a solution of the infinite-dimensional Navier–Stokes equations. The Countable Choice assumption is inherited from the periodic Fourier-Sobolev framework; the finite-dimensional proof makes no further choice.

Statement references

Finite solenoidal Galerkin coefficient dynamics

Picard–Lindelöf local existence and uniqueness

Wiedemann, §3.1.2 and §3.2, Theorem 3.4 proof Steps 1–2

Ryzhik, §5, equations (5.24)–(5.35)

Facts & Assumptions

Given: The finite smooth solenoidal orthonormal family, normalized periodic measure, parameter $ν \geq 0$ , initial vector $a^{0}$ , and the coefficient system in the Statement.

[F1]

Finite solenoidal Galerkin coefficient dynamics: The classes lie in the supplied $H^{0} (Q_{d}; R^{d})$ Fourier completion with unique normalized- $L^{2}$ representatives; the chosen representatives are smooth, periodic, divergence-free, and orthonormal, and the coefficient integrals and ODE are defined from them. The supplier assumes Countable Choice.

[F2]

Picard-Lindelöf local existence and uniqueness for first-order systems: A continuous vector field locally Lipschitz in the state variable on an open ODE domain has a unique local solution through each initial point.

[F3]

Picard-Lindelöf local existence and uniqueness for first-order systems: Any two solutions through the same initial data agree on every common subinterval containing the initial time.

[F4]

$R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ : For every natural number $n \geq 1$ , the Euclidean metric space $R^{n}$ is complete. Here $r \in N$ and $r \geq 1$ , so this applies to $R^{r}$ .

Proof

1.1givenF1F2algebra

Define $F : R^{r} \to R^{r}$ by $F_{i} (a) = - \sum_{j, k} c_{i j k} a_{j} a_{k} - ν \sum_{j} A_{i j} a_{j}$ , and lift it to the time-state map $\tilde{F} : R \times R^{r} \to R^{r}$ , $\tilde{F} (t, a) = F (a)$ , on the open domain $D = R \times R^{r}$ . This lift is continuous. For any compact time-state cylinder $K \subset D$ , its state projection lies in a closed ball ${\overline{B}}_{R}$ ; since $F$ is polynomial, $\sup_{a \in {\overline{B}}_{R}} ∥ D F (a) ∥_{o p} < \infty$ . The mean-value estimate along line segments in ${\overline{B}}_{R}$ therefore gives one state-Lipschitz constant for all $(t, a), (t, b) \in K$ with the same $t$ ; the constant is independent of $t$ because $\tilde{F}$ does not depend on time. Thus $\tilde{F}$ is locally Lipschitz in the state variable, locally uniformly in time, and [F2] applies on $D$ .

2.1givenF2F3step 1.1construct

Consider all solutions through $(0, a^{0})$ on open intervals containing $0$ . This collection is nonempty by [F2]. Any two such solutions agree on the intersection of their intervals by [F3]; hence their interval union $I$ is an open interval and their common values define one solution $a : I \to R^{r}$ . Every solution through the same data is a restriction of this union solution.

2.2givenF1step 1.1assume-contraalgebra

For each $i, j, k$ , periodic integration by parts and $div v_{j} = 0$ give $c_{i j k} + c_{k j i} = \int_{Q_{d}} v_{j} \cdot \nabla (v_{k} \cdot v_{i}) d μ = \int_{Q_{d}} div (v_{j} (v_{k} \cdot v_{i})) d μ = 0$ , since opposite faces of a periodic cell cancel. Thus $c_{k j i} = - c_{i j k}$ ; the finite contraction $S = \sum_{i, j, k} c_{i j k} a_{i} a_{j} a_{k}$ satisfies $S = - S$ after swapping $i$ and $k$ , so $S = 0$ . Also $\sum_{i, j} A_{i j} a_{i} a_{j} = \int_{Q_{d}} ∣ \nabla u_{a} ∣^{2} d μ$ . Multiplying the coefficient equations by $a_{i}$ , summing over $i$ , and using $\frac{d}{d t} \frac{1}{2} ∣ a ∣^{2} = \sum_{i} a_{i} a_{i}^{'}$ proves the energy identity on $I$ . Suppose for contradiction that the positive endpoint $β : = \sup (I \cap (0, \infty))$ is finite.

3.1givenstep 2.2algebra

Integrating the identity from $0$ to any $t \in I \cap [0, \infty)$ gives $∣ a (t) ∣^{2} + 2 ν \int_{0}^{t} \int_{Q_{d}} ∣ \nabla u_{a (s)} ∣^{2} d μ d s = ∣ a^{0} ∣^{2}$ . Because $ν \geq 0$ , this implies $∣ a (t) ∣ \leq R : = ∣ a^{0} ∣$ for every $0 \leq t < β$ .

4.1givenF4step 1.1step 3.1algebra

On $∣ a ∣ \leq R$ , each component of $F$ obeys $∣ F_{i} (a) ∣ \leq R^{2} \sum_{j, k} ∣ c_{i j k} ∣ + ν R \sum_{j} ∣ A_{i j} ∣$ , so $∣ F (a) ∣ \leq M$ for the finite constant $M = (\sum_{i} (R^{2} \sum_{j, k} ∣ c_{i j k} ∣ + ν R \sum_{j} ∣ A_{i j} ∣)^{2})^{1 / 2}$ . Therefore $∣ a (t) - a (s) ∣ \leq M ∣ t - s ∣$ for $0 \leq s < t < β$ by integrating $a^{'} = F (a)$ . Set $t_{n} = β (1 - 1 / (n + 2))$ for $n \in N$ ; then $t_{n} ↑ β$ and $(a (t_{n}))$ is Cauchy by this bound, so [F4] gives a limit $a_{*} \in R^{r}$ . For $t_{n} \leq t < β$ , the same bound gives $∣ a (t) - a_{*} ∣ \leq M (β - t_{n}) + ∣ a (t_{n}) - a_{*} ∣ \to 0$ as $n \to \infty$ , so $a (t) \to a_{*}$ as $t ↑ β$ .

5.1F2step 2.1step 4.1discharge-contradiction∎

Extend $a$ continuously by $a (β) = a_{*}$ . For $h < 0$ sufficiently close to zero, the quotient $(a (β + h) - a_{*}) / h = ∣ h ∣^{- 1} \int_{β + h}^{β} \tilde{F} (s, a (s)) d s$ tends to $\tilde{F} (β, a_{*})$ as $h ↑ 0$ . Applying [F2] to the same lift $\tilde{F}$ at $(β, a_{*})$ gives a local solution $b$ on an interval around $β$ with $b (β) = a_{*}$ and $b^{'} (t) = \tilde{F} (t, b (t))$ . Define the joined curve to equal $a$ before $β$ and $b$ at and after $β$ . Its values and one-sided derivatives match at $β$ , and continuity of $\tilde{F}$ makes its derivative continuous there, so it is a solution on a strictly larger open interval. This contradicts the construction of $I$ as the union of all solution intervals. Hence $β = \infty$ , and uniqueness follows from the overlap agreement in step 2.1.

Skew reduced convection and global finite dynamics

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources