LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Positive-semidefinite entropy-variable factorization of viscous flux

Statement

For $U \in G$ , let $Z = (Z_{1}, \dots, Z_{d})$ and $W = (W_{1}, \dots, W_{d})$ be arrays of entropy-variable gradients, with $Z_{k}, W_{k} \in R^{d + 2}$ . Write $Z_{k, m_{ℓ}}$ for the momentum-variable component and $Z_{k, E}$ for the energy-variable component. Define

$G_{k ℓ} (Z) = Z_{k, m_{ℓ}} + u_{ℓ} Z_{k, E}, c_{k} (Z) = Z_{k, E},$

and $G^{0} (Z) = sym G (Z) - (tr G (Z) / d) I$ . Define the linear map

$R_{U} Z = (\sqrt{2 μ T} G^{0} (Z), \sqrt{ζ T} tr G (Z), \sqrt{κ} T c (Z)) .$

The bilinear form

$B_{U} (Z, W) = 2 μ T G^{0} (Z) : G^{0} (W) + ζ T tr G (Z) tr G (W) + κ T^{2} \sum_{k = 1}^{d} c_{k} (Z) c_{k} (W)$

has the factorization $B_{U} (Z, W) = (R_{U} Z) \cdot (R_{U} W) = \sum_{j, k} Z_{j}^{T} K_{j k} (U) W_{k}$ , so $K (U) = R_{U}^{*} R_{U}$ is a symmetric positive-semidefinite block matrix. For $Z_{k} = \partial_{k} v (U)$ , its blocks give exactly the model viscous flux $F_{j}^{v i s c} = \sum_{k} K_{j k} (U) \partial_{k} v$ , and

$\sum_{j, k} (\partial_{j} v)^{T} K_{j k} (U) (\partial_{k} v) = \frac{τ : \nabla u}{T} + \frac{κ ∣ \nabla T ∣^{2}}{T^{2}} \geq 0.$

Facts & Assumptions

Given: $U \in G$ , the constants $μ > 0$ , $ζ \geq 0$ , $κ > 0$ , and arbitrary arrays $Z, W \in (R^{d + 2})^{d}$ .

[F1]

Ideal-gas compressible Navier–Stokes–Fourier model: $D = sym \nabla u$ , $D^{0} = D - (tr D / d) I$ , $τ = 2 μ D^{0} + ζ (tr D) I$ , and $F_{j}^{v i s c} = (0, τ_{j 1}, \dots, τ_{j d}, u \cdot τ_{j \cdot} + κ \partial_{j} T)$ .

[F2]

Strict convexity and entropy variables for the ideal-gas Euler flux: $v_{m} = β u$ , $v_{E} = - β$ , and $β = 1 / T$ .

Proof

1.1F2algebra

For any entropy-variable gradient array $Z$ , set $G_{k ℓ} = Z_{k, m_{ℓ}} + u_{ℓ} Z_{k, E}$ and $c_{k} = Z_{k, E}$ . The formula for $v$ in [F2] gives, when $Z_{k} = \partial_{k} v$ , $G_{k ℓ} = T^{- 1} \partial_{k} u_{ℓ}$ and $c_{k} = T^{- 2} \partial_{k} T$ , since $\partial_{k} (β u_{ℓ}) + u_{ℓ} \partial_{k} (- β) = β \partial_{k} u_{ℓ}$ and $\partial_{k} (- 1 / T) = T^{- 2} \partial_{k} T$ .

2.1F1F2step 1.1algebra

For a general array $Z$ , define $τ_{k ℓ} (Z) = 2 μ T G^{0} (Z)_{k ℓ} + ζ T tr G (Z) δ_{k ℓ}$ . The displayed bilinear form expands to $B_{U} (Z, W) = \sum_{k, ℓ} (W_{k, m_{ℓ}} + u_{ℓ} W_{k, E}) τ_{k ℓ} (Z) + κ T^{2} \sum_{k} W_{k, E} Z_{k, E}$ . Thus the linear functional in $W$ is pairing against the flux with zero mass row, momentum entries $τ_{k ℓ} (Z)$ , and energy entry $u \cdot τ_{k \cdot} (Z) + κ T^{2} Z_{k, E}$ . By step 1.1 this is precisely the physical viscous flux when $Z = \nabla v$ .

3.1F1step 2.1algebra

The linear map $R_{U}$ in the Statement gives $B_{U} (Z, W) = (R_{U} Z) \cdot (R_{U} W)$ term by term. Its adjoint therefore defines the block matrix $K (U) = R_{U}^{*} R_{U}$ , which is symmetric and positive semidefinite because $Z^{T} K (U) Z = ∣ R_{U} Z ∣^{2} \geq 0$ . In particular, for $W = Z$ the quadratic form equals $2 μ T ∣ G^{0} (Z) ∣^{2} + ζ T (tr G (Z))^{2} + κ T^{2} \sum_{k} Z_{k, E}^{2} \geq 0$ , using $T > 0$ , $μ > 0$ , $ζ \geq 0$ , and $κ > 0$ .

4.1F1F2step 1.1step 3.1algebra∎

Take $Z = \nabla v$ . Then $\nabla u = T G (Z)$ and $\nabla T = T^{2} c (Z)$ by step 1.1. Since $τ$ is symmetric, $τ : \nabla u = 2 μ T^{2} ∣ G^{0} (Z) ∣^{2} + ζ T^{2} (tr G (Z))^{2}$ . Dividing by $T$ and adding $κ ∣ \nabla T ∣^{2} / T^{2} = κ T^{2} ∣ c (Z) ∣^{2}$ gives exactly $B_{U} (Z, Z)$ , proving the production identity and its nonnegativity.

Depends on

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pablo Fernandez, Ngoc-Cuong Nguyen, and Jaime Peraire, Entropy-stable hybridized discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations, complete author preprint (standard reference, not scraped)
Eitan Tadmor, Entropy Stable Schemes, Chapter 18, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, 2016, complete chapter (standard reference, not scraped)