ExampleConstruction: AI-generatedVerification: 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.

Conditional nonlinear Couette energy stability — worked interpretation

Physical scope

Exact single-mode smooth solution in a periodic channel with Couette wall velocities. It verifies one conditional energy-decay instance and does not imply global existence, an optimal threshold, or an observed transition value.

Example

Assume ${A C}_{ω}$ . Take SI values $h = 1$ metre, $L_{x} = L_{z} = 2 π$ metres, $S = 1$ inverse second, $ρ_{0} = 1$ kilogram per cubic metre, and $μ = 1$ pascal-second, so $ν = μ / ρ_{0} = 1$ square metre per second. Let $T > 0$ seconds and choose an amplitude $a \neq 0$ metres per second. For a length $L > 0$ , define $T_{L} = R / (L Z)$ , with real coordinates measured in metres; here both horizontal periods are $L = L_{x} = L_{z} = 2 π m$ . On $Q = T_{2 π m} \times (0, 1 m) \times T_{2 π m}$ , let $p_{0} \in R$ be an arbitrary constant pressure measured in pascals, and set $p = p_{0}$ and

$u (x, y, z, t) = (S y + a e^{- ν π^{2} t / h^{2}} \sin ⁣ (\frac{π y}{h}), 0, 0), 0 \leq t \leq T .$

The field is periodic in $x, z$ , divergence-free, and has Couette wall values $0$ at $y = 0$ and $S h (1, 0, 0)$ at $y = h$ . Its perturbation energy is $E (t) = E (0) e^{- 2 ν π^{2} t / h^{2}}$ . Since $∣ S ∣ h^{2} / ν = 1 < 2$ , this exact decay is stronger than the sufficient bound $E (t) \leq e^{- (2 ν / h^{2} - ∣ S ∣) t} E (0)$ for every $t \in [0, T]$ .

Facts & Assumptions

Given: The SI parameters, periodic channel, constant pressure, amplitude, and velocity field in the Example.

[F1]

Classical frames, primitive quantities and SI units: time is in seconds, length in metres, velocity in metres per second, and the selected Newtonian viscosity and density have units Pa s and kg m $^{- 3}$ .

[F2]

Specified wall, thermal, periodic and initial data: the chosen no-slip data match the wall velocity at both channel walls; the horizontal cell faces are periodic identifications.

[F3]

Conditional nonlinear Couette energy stability — conditional physical prediction: an already-existing smooth solution with these periodic Couette data obeys the stated sufficient conditional energy estimate when $∣ S ∣ h^{2} / ν < 2$ .

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ means every sequence of nonempty sets has a choice function; the example carries the exact mathematical assumption of [F3].

[A1]

All stated parameters, including the arbitrary constant pressure $p_{0}$ , are selected model inputs, not measured values. The example is a constructed exact solution and supplies no empirical observation.

Verification

1.1givenF1F2algebra

The velocity depends only on $y, t$ in its first component, so $\nabla \cdot u = 0$ and $(u \cdot \nabla) u = 0$ . Writing $w_{x} = a e^{- ν π^{2} t / h^{2}} \sin (π y / h)$ gives $\partial_{t} w_{x} = - (ν π^{2} / h^{2}) w_{x}$ and $\partial_{y y} w_{x} = - (π^{2} / h^{2}) w_{x}$ ; the Couette base $S y$ has zero time derivative and Laplacian. Hence $\partial_{t} u + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ . The sine term vanishes at both walls, and the remaining values are $0$ and $S h$ , so the no-slip boundary conditions hold. The field is smooth on the finite interval and constant, hence periodic, in $x, z$ .

2.1givenstep 1.1algebra

The perturbation is $w = (a e^{- ν π^{2} t / h^{2}} \sin (π y / h), 0, 0)$ . Integrating over the fundamental cell and using $\int_{0}^{h} \sin^{2} (π y / h) d y = h / 2$ gives $E (t) = L_{x} L_{z} (a^{2} h / 2) e^{- 2 ν π^{2} t / h^{2}} = E (0) e^{- 2 ν π^{2} t / h^{2}}$ .

3.1givenF3F4step 2.1algebra∎

The selected parameters give $∣ S ∣ h^{2} / ν = 1 < 2$ and $2 ν / h^{2} - ∣ S ∣ = 1 s^{- 1}$ . Since $2 π^{2} > 1$ , for $t \geq 0$ we have $e^{- 2 ν π^{2} t / h^{2}} \leq e^{- (2 ν / h^{2} - ∣ S ∣) t}$ . Thus the computed energy satisfies the bound of [F3] throughout $[0, T]$ . This is one model calculation, not a global-existence or experimental claim.

Conditional nonlinear Couette energy stability — worked interpretation

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels