CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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Equal means with unequal Reynolds covariance

Statement refuted

Fix any integer $d \geq 2$ and any $ν \geq 0$ , and define $T^{d} = (R / 2 π Z)^{d}$ . Consider the entire class of finite ensembles described as follows, allowing the number of members and the weight tuple to vary across the class. Each ensemble has an integer $N \geq 1$ and fixed weights $a_{j} \geq 0$ with $\sum_{j = 1}^{N} a_{j} = 1$ . For $j = 1, \dots, N$ , let $u_{j} : R \times R^{d} \to R^{d}$ be continuously differentiable in $(t, x)$ with continuous second spatial derivatives, and let $p_{j} : R \times R^{d} \to R$ be continuously differentiable. Require all these fields to be $2 π$ -periodic in each spatial coordinate, and suppose that for each $j = 1, \dots, N$ , pointwise on $R \times R^{d}$ ,

$div u_{j} = 0, \partial_{t} u_{j} + div (u_{j} \otimes u_{j}) + \nabla p_{j} = ν Δ u_{j} .$

For each such family, define $\overset{ˉ}{u} = \sum_{j} a_{j} u_{j}$ , $\overset{ˉ}{p} = \sum_{j} a_{j} p_{j}$ , and $R = \sum_{j} a_{j} (u_{j} - \overset{ˉ}{u}) \otimes (u_{j} - \overset{ˉ}{u})$ . The following universal determination assertion over this entire class is false: any two ensembles in the class with the same mean velocity and mean pressure fields $\overset{ˉ}{u}, \overset{ˉ}{p}$ must have the same covariance tensor $R$ .

Statement references

Flierl and Ferrari, MIT OCW 12.820, Chapter 3, §3.6 (closure problem), full lecture notes

Facts & Assumptions

[F1]

Exact finite Reynolds averaged equation: For integers $d, N \geq 1$ , an open time interval $I$ , an open spatial domain $Ω \subseteq R^{d}$ , fixed nonnegative weights summing to one, and common $ν \geq 0$ , the finite Reynolds theorem assumes velocities $u_{j} : I \times Ω \to R^{d}$ continuously differentiable in $(t, x)$ with continuous second spatial derivatives, continuously differentiable pressures $p_{j} : I \times Ω \to R$ , continuous forcings $f_{j} : I \times Ω \to R^{d}$ , and pointwise equations $div u_{j} = 0$ and $\partial_{t} u_{j} + div (u_{j} \otimes u_{j}) + \nabla p_{j} = ν Δ u_{j} + f_{j}$ . It defines $\overset{ˉ}{u} = \sum_{j} a_{j} u_{j}$ , $v_{j} = u_{j} - \overset{ˉ}{u}$ , and $R = \sum_{j} a_{j} v_{j} \otimes v_{j}$ , and proves the exact averaged equation with covariance term $- div R$ . In this counterexample $I = R$ , $Ω = R^{d}$ , and $f_{j} = 0$ .

[A1]

Fix any integer $d \geq 2$ , any $ν \geq 0$ , and any nonzero constant vector $w \in R^{d}$ .

[A2]

The two proposed ensembles each have two members (so $N = 2$ ) and fixed weights $a_{1} = a_{2} = 1 / 2$ ; all velocities and pressures are the constant fields specified in the Counterexample steps.

Given: The torus, common viscosity, finite Reynolds equation and covariance convention [F1], and the nonzero vector [A1].

Counterexample

1.1givenA1A2algebra

Define ensemble A by $u_{1} = u_{2} = 0$ and $p_{1} = p_{2} = 0$ . Define ensemble B by $u_{1} = w$ , $u_{2} = - w$ , and $p_{1} = p_{2} = 0$ , with weights $1 / 2, 1 / 2$ in each. These velocity and pressure fields are constant, so the velocities are continuously differentiable with continuous second spatial derivatives, the pressures are continuously differentiable, and all fields are periodic. Every time and spatial derivative vanishes, so every member is divergence-free and satisfies the displayed equation for the common $ν$ . The mean velocities in both ensembles are $0$ , and the mean pressures in both are $0$ .

2.1givenF1A1A2step 1.1algebra

By [F1], the covariance tensor in the averaged equation is $R = \sum_{j} a_{j} (u_{j} - \overset{ˉ}{u}) \otimes (u_{j} - \overset{ˉ}{u})$ . For ensemble A this is $R_{A} = 0$ . For ensemble B the fluctuations are $w, - w$ , hence $R_{B} = \frac{1}{2} w \otimes w + \frac{1}{2} (- w) \otimes (- w) = w \otimes w \neq 0$ , since $w^{T} R_{B} w = ∣ w ∣^{4} > 0$ .

3.1step 1.1step 2.1algebra∎

The two admissible ensembles have identical mean velocity and pressure fields but unequal covariance tensors, contradicting the refuted universal determination claim. This excludes only unrestricted determination from those means; it does not address a closure relation restricted by additional hypotheses.

Depends on

Exact finite Reynolds averaged equation

Dependency tree · 2 levels

Sources

Glenn Flierl and Raffaele Ferrari, MIT OpenCourseWare 12.820, Turbulence in the Ocean and Atmosphere, Chapter 3, §3.6 (Spring 2006 lecture notes)