ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Exact constant-density compressible viscous shear

Example

Assume Countable Choice. Let $ℓ, ρ_{*}, η, κ > 0$ , $γ > 1$ , $ζ \geq 0$ , and $A \in R$ . On $T_{ℓ}^{3} = (R / 2 π ℓ Z)^{3}$ with coordinates $(x, y, z)$ , set $ν = η / ρ_{*}$ , $p_{*} = κ ρ_{*}^{γ}$ , and $V (t, y) = A e^{- ν t / ℓ^{2}} \sin (y / ℓ)$ . Then $ρ = ρ_{*}$ and $u = (V, 0, 0)$ give a global smooth solution with no external force of $ρ_{t} + div (ρ u) = 0, (ρ u)_{t} + div (ρ u \otimes u) + \nabla (κ ρ^{γ}) = div ⁣ (2 η D_{0} + ζ (div u) I),$ where $D_{0} = (\nabla u + \nabla u^{T}) / 2 - (div u) I / 3$ . This is the first sine heat mode of the compatible channel shear evolution at channel height $h = π ℓ$ , periodically extended in $y$ . It is one exact solution of the compressible viscous system, not a general nonconstant-density existence result. With SI units, $x, y, z, ℓ$ are metres, $t$ is seconds, $A, u$ are m/s, $ρ_{*}$ is kg/m $^{3}$ , $p_{*}$ is Pa, $η, ζ$ are Pa $\cdot$ s, $κ$ is Pa $\cdot$ (m $^{3}$ /kg) $^{γ}$ , and $ν$ is m $^{2}$ /s.

Example references

Global compatible no-slip shear evolution

The Axiom of Countable Choice ( ${A C}_{ω}$ )

David Tong, Fluid Mechanics, Chapter 2, §§3.2.1–3.2.3

Verification

Given: Countable Choice, the torus, constants, constitutive stress, and compressible equations in the Example.

[F1] For channel height $h = π ℓ$ , the sine-mode heat evolution has terms $b_{n} e^{- ν (n π / h)^{2} t} \sin (n π y / h)$ and is smooth for compatible odd periodic data (Global compatible no-slip shear evolution).

[A1] Countable Choice is carried from the compatible shear theorem's Fourier-basis input (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2] The conservative mass and momentum equations and Newtonian stress tensor displayed in the Example are the mathematical system being checked; no empirical constitutive assertion is made.

1.1givenF1A1algebra

Take $F (y) = A \sin (y / ℓ)$ on $0 \leq y \leq π ℓ$ . Its odd extension has period $2 π ℓ$ and is smooth. Under Countable Choice [A1], the channel Fourier formula [F1] has all sine coefficients zero except $b_{1} = A$ , because $2 / (π ℓ) \int_{0}^{π ℓ} A \sin^{2} (y / ℓ) d y = A$ . The resulting $V = A e^{- ν t / ℓ^{2}} \sin (y / ℓ)$ is smooth and $2 π ℓ$ -periodic in $y$ , so it defines the stated torus field and satisfies $V_{t} = ν V_{y y}$ .

2.1givenstep 1.1A2algebra

Since $u = (V (y), 0, 0)$ is independent of $x$ and $ρ$ is constant, $ρ_{t} = 0$ and $div (ρ u) = ρ_{*} \partial_{x} V = 0$ . Thus the mass equation in [A2] holds.

3.1givenstep 1.1A2algebra∎

We have $div u = 0$ , $u \cdot \nabla u = V \partial_{x} u = 0$ , and $\nabla (κ ρ_{*}^{γ}) = 0$ . The symmetric gradient has only $x y$ and $y x$ entries, each $V_{y} / 2$ , so it is trace-free and $D_{0} = D$ . Therefore $div (2 η D_{0}) = η Δ u = (η V_{y y}, 0, 0)$ and the bulk-viscosity term is zero. The momentum left side in [A2] is $(ρ_{*} V_{t}, 0, 0) = (η V_{y y}, 0, 0)$ by step 1.1 and $ν = η / ρ_{*}$ , so the momentum equation holds. This verifies the exact solution for all $t \geq 0$ .

Depends on

Dependency tree · 2 levels

Sources

David Tong, Fluid Mechanics, Chapter 2, §§3.2.1–3.2.3