Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Mass density reconstruction and incompressibility

Physical scope

Conditional on the adopted smooth single-component mass balance, a C2 orientation-preserving material motion with positive Jacobian, and a C1 positive density. Vacuum, shocks, weak solutions, non-injective motions, and empirical validity are excluded.

Statement

Let $I \subseteq R$ be an open time interval containing $0$ , let $B \subseteq R^{3}$ be a bounded $C^{1}$ reference domain, and let $χ \in C^{2} (I \times U, R^{3})$ be an orientation-preserving family of diffeomorphisms on an open neighborhood $U \supseteq \overline{B}$ , with $χ (0, a) = a$ and $J (t, a) = \det D_{a} χ (t, a) > 0$ . Set $u (t, χ (t, a)) = \partial_{t} χ (t, a)$ . Let the current density $ρ \in C^{1}$ be positive on the swept material region, and define $ρ_{0} (a) = ρ (0, a)$ . Assume the adopted single-component mass balance with no sources or diffusion relative to $u$ , on every bounded $C^{1}$ material subregion. Then this integral material-mass balance is equivalent, under these smoothness hypotheses, to $\partial_{t} ρ + div (ρ u) = 0$ . It also forces $ρ (t, χ (t, a)) J (t, a) = ρ_{0} (a)$ . Consequently, for positive density, $D_{t} ρ = 0$ throughout the material region if and only if $div u = 0$ there. This is a conditional deduction within the adopted smooth model; it asserts no result for vacuum, shocks, weak solutions, or empirical applicability.

Statement references

Ain A. Sonin, MIT 2.25 lecture notes, §§1.1–1.3 (material mass balance)

Andrew Hazel, MATH45061 Continuum Mechanics, Chapter 4, §4.1 (Lagrangian and Eulerian mass conservation)

Facts & Assumptions

Given: The stated C2 orientation-preserving motion, its positive Jacobian, a C1 positive current density, the reference normalization $χ (0, a) = a$ , and the adopted no-mass-source material balance.

[A1]

Single-component nonpolar continuum and material balances: for each bounded $C^{1}$ material subregion, its total mass is constant; the adopted branch has no mass sources or diffusion relative to $u$ .

[F1]

Smooth transport, Jacobian and advected reference density: for a $C^{2}$ orientation-preserving motion and a $C^{1}$ scalar field $ρ$ on the swept open set, every compact Jordan material set $K$ has $M_{K} (t) = \int_{χ_{t} (K)} ρ (t, x) d x$ with $M_{K}^{'} (t) = \int_{χ_{t} (K)} (\partial_{t} ρ + div (ρ u)) d x .$ The same statement supplies the change-of-variables formula for $χ_{t} (K)$ , the identity $\frac{d}{d t} [ρ (t, χ (t, a)) J (t, a)] = J (D_{t} ρ + ρ div u) (t, χ (t, a))$ , and the equivalence between continuity, constancy of all compact-Jordan material integrals, and invariance of the density-Jacobian product. This remains a draft mathematical supplier, and the proof uses its stated interface as a mathematical premise.

[F2]

Classical frames, primitive quantities and SI units: $u$ is the Eulerian velocity, $D_{t} = \partial_{t} + u \cdot \nabla$ is the material derivative, $θ = div u$ , and $div (ρ u)$ is the spatial divergence of the mass flux.

[F3]

C1 material regions have compatible Jordan volume integrals: for a bounded open region with the stated local $C^{1}$ graph boundary, its compact closure and its image under a $C^{1}$ diffeomorphism are Jordan measurable; the boundaries have content zero, and integration over the open current region equals integration over its compact closure.

[F4]

Continuous residuals localize on C1 domains: for a continuous scalar residual, zero integrals on every bounded $C^{1}$ subdomain force pointwise zero, and nonnegative integrals on all such subdomains force pointwise nonnegativity.

Proof

Given: [A1], [F1], the stated smooth motion and density, and $J > 0$ .

1.1A1F3

For each bounded $C^{1}$ subdomain $V ⋐ B$ , put $K = \overline{V}$ . The adopted mass balance [A1] makes $\int_{χ_{t} (V)} ρ$ constant in time. The Jordan-interface lemma [F3] identifies this with $M_{K} (t) = \int_{χ_{t} (K)} ρ$ , so $M_{K}^{'} (t) = 0$ .

1.2F1F3given

Conversely, suppose $R : = \partial_{t} ρ + div (ρ u) = 0$ . Then [F1] makes $M_{K}$ constant for every compact Jordan $K$ . For $K = \overline{V}$ and each bounded $C^{1}$ subdomain $V ⋐ B$ , [F3] identifies the open-region mass integral with $M_{K}$ ; its value at $t = 0$ is $\int_{V} ρ_{0}$ . Thus the adopted integral balance follows, proving the reverse implication.

2.1F1F3step 1.1

Apply [F1] to $q = ρ$ and $K$ from step 1.1. Since $M_{K}^{'} = 0$ , its material-transport formula and change of variables give $\int_{V} R (t, χ (t, a)) J (t, a) d a = 0, R : = \partial_{t} ρ + div (ρ u) .$ Here [F3] replaces the integral over the compact closure by the equal integral over the open region $V$ . This holds for every bounded $C^{1}$ subdomain $V ⋐ B$ .

3.1F4step 2.1algebra

For fixed $t$ , the function $a \mapsto R (t, χ (t, a)) J (t, a)$ is continuous on $B$ . Step 2.1 and [F4] therefore imply that it vanishes throughout $B$ . Since $J > 0$ , the local continuity equation $R = 0$ holds on the swept material region.

4.1F1F2step 3.1algebra∎

Along each material label, [F1] gives $\frac{d}{d t} (ρ (t, χ (t, a)) J (t, a)) = J (D_{t} ρ + ρ div u)$ . By [F2] and step 3.1 this derivative vanishes. The initial conditions $χ (0, a) = a$ , $J (0, a) = 1$ , and $ρ (0, a) = ρ_{0} (a)$ give $ρ (t, χ (t, a)) J (t, a) = ρ_{0} (a)$ . Since $ρ > 0$ , $D_{t} ρ + ρ div u = 0$ is equivalent to $D_{t} ρ = 0$ exactly when $div u = 0$ .

Authoring status

The mathematical motion and transport carriers are present and their complete statements/proofs were checked against this use. The transport lemma remains a draft mathematical supplier, and the proof's reliance on its stated interface is explicit. The open-to-compact Jordan conversion and continuous localization are supplied by C1 material regions have compatible Jordan volume integrals and Continuous residuals localize on C1 domains.

Mass density reconstruction and incompressibility

Statement

Statement references

Facts & Assumptions

Proof

Authoring status

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