TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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Smooth polytropic energy conservation

Statement

Assume Countable Choice. Let $d \in {1, 2, 3}$ , $κ > 0$ , $γ > 1$ , $a = (γ - 1) / 2$ , integer $s \geq 4$ , and smooth real periodic data $ρ_{0} : Q_{d} \to (0, \infty)$ and $u_{0} : Q_{d} \to R^{d}$ on the normalized torus $Q_{d} = R^{d} / (2 π Z)^{d}$ , with $\min_{x \in Q_{d}} ρ_{0} (x) > 0$ . Use normalized Lebesgue measure $d μ = (2 π)^{- d} d x$ , Fourier coefficients $\hat{f} (k) = \int_{Q_{d}} f (x) e^{- i k \cdot x} d μ (x)$ , and norms $∥ f ∥_{H^{m}}^{2} = \sum_{k \in Z^{d}} (1 + ∣ k ∣^{2})^{m} ∣ \hat{f} (k) ∣^{2}$ . For $ρ > 0$ , set $r (ρ) = \sqrt{κ γ} ρ^{a} / a$ . The local positive-density polytropic Euler theorem supplies a unique smooth solution $(ρ, u)$ on $[0, T]$ with these initial data and $ρ (t, x) > 0$ , where $T$ depends only on $d, s, κ, γ, ∥ (r (ρ_{0}), u_{0}) ∥_{H^{s}}$ and $\min_{x \in Q_{d}} r (ρ_{0}) (x)$ , satisfying $ρ_{t} + div (ρ u) = 0, u_{t} + u \cdot \nabla u + ρ^{- 1} \nabla p (ρ) = 0, p (ρ) = κ ρ^{γ} .$ Define the internal-energy density $U (ρ) = κ ρ^{γ} / (γ - 1)$ and total-energy density $E = \frac{1}{2} ρ ∣ u ∣^{2} + U (ρ)$ . Then $\partial_{t} E + div ((E + p (ρ)) u) = 0,$ and the total energy $\int_{Q_{d}} E (t, x) d μ (x)$ is constant on $[0, T]$ .

Statement references

Sound-speed variables for polytropic Euler

Local smooth positive-density polytropic Euler

Tong, Fluid Mechanics, Chapter 3, §4.5

Golse, The Boltzmann Equation and Its Hydrodynamic Limits, Chapter 3

Facts & Assumptions

Given: Countable Choice, $κ > 0$ , $γ > 1$ , and the smooth positive-density solution in the Statement.

[A1]

Countable Choice: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Sound-speed variables for polytropic Euler: $p (ρ) = κ ρ^{γ}$ for $ρ > 0$ and the admissible sound-speed variable is defined only on positive density.

[F2]

Local smooth positive-density polytropic Euler: for the stated positive-density periodic data and parameters, the polytropic Euler system has a unique local smooth solution; its present proof uses the exact FD-S01 Fourier estimates conditionally.

Proof

1.1givenA1F2algebra

By [F2], the unique smooth positive-density solution exists on its local lifespan. For every smooth scalar $ϕ$ , the continuity equation and product rule give $\partial_{t} (ρ ϕ) + div (ρ u ϕ) = ρ (\partial_{t} ϕ + u \cdot \nabla ϕ)$ . Indeed, expansion of the left side produces $ϕ (ρ_{t} + div (ρ u)) + ρ ϕ_{t} + ρ u \cdot \nabla ϕ$ , and the first term is zero.

1.2givenF1algebra

For $U (ρ) = κ ρ^{γ} / (γ - 1)$ , the continuity equation gives $\partial_{t} U + div (U u) = U^{'} (ρ) (ρ_{t} + u \cdot \nabla ρ) + U div u = (U - ρ U^{'}) div u$ . Direct differentiation gives $U - ρ U^{'} = - κ ρ^{γ} = - p$ , hence $\partial_{t} U + div (U u) = - p div u$ .

2.1givenstep 1.1algebra

Put $K = ρ ∣ u ∣^{2} / 2$ . The momentum equation is $ρ (\partial_{t} u + u \cdot \nabla u) = - \nabla p$ . Apply step 1.1 to $ϕ = ∣ u ∣^{2} / 2$ , so $\partial_{t} K + div (K u) = ρ u \cdot (\partial_{t} u + u \cdot \nabla u) = - u \cdot \nabla p$ .

3.1givenstep 2.1step 1.2algebra∎

Add steps 2.1 and 1.2 and use $u \cdot \nabla p + p div u = div (p u)$ . This yields $\partial_{t} (K + U) + div ((K + U + p) u) = 0$ , which is the asserted local conservation law. Integrating it over the periodic torus makes the divergence integral vanish, so the total energy is constant.

Smooth polytropic energy conservation

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources