TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck pendingverified 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.

Smooth rotating shallow-water energy balance

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Let $I \subseteq R$ be a nonempty open time interval and use $T^{2} = R^{2} / (2 π Z)^{2}$ with Lebesgue measure $λ_{2}$ on $Q_{2} = [0, 2 π]^{2}$ , where $d x = d λ_{2} (x)$ as defined for the equation family. Let $h : I \times T^{2} \to (0, \infty)$ and $v : I \times T^{2} \to R^{2}$ be continuously differentiable in time and space with continuous second spatial derivatives and $2 π$ -periodic lifts. Suppose they satisfy the one-layer rotating shallow-water equations with constant $f \in R$ , constant $g > 0$ , fixed flat bottom, and no friction:

$h_{t} + \nabla \cdot (h v) = 0, (h v)_{t} + \nabla \cdot (h v \otimes v) + \nabla ⁣ (\frac{g h^{2}}{2}) + f h J v = 0,$

where $J (v_{1}, v_{2}) = (- v_{2}, v_{1})$ . Define the normalized energy density per horizontal area by $E : = h ∣ v ∣^{2} / 2 + g h^{2} / 2$ ; it is per unit reference density, so a specified constant density converts its area integral to joules. Then the pointwise energy flux identity is

$\partial_{t} E + \nabla \cdot ((E + \frac{g h^{2}}{2}) v) = 0.$

Consequently, for every $t_{0} < t_{1}$ in $I$ ,

$\int_{Q_{2}} E (t_{1}, x) d x = \int_{Q_{2}} E (t_{0}, x) d x .$

This is an exact smooth identity for the stated adopted equation system, not an aspect-ratio or hydrostatic-error estimate and not a positivity guarantee for an arbitrary discretization. Bottom topography, wetting/drying, bottom friction, or multiple layers require separately defined systems and their corresponding energy terms.

Statement references

Equation family and conventions: Rotating stratified and shallow-water equation families.

Continuous functions on closed rectangles are Riemann integrable: Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable.

Under ${A C}_{ω}$ , bounded Borel Riemann integrands on finite-dimensional boxes have the same Lebesgue and Riemann integrals: Borel Darboux integrands in finite dimension.

Riemann integration over products of rectangles: Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections.

Newton–Leibniz for continuous endpoint functions with an integrable interior derivative: Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative.

Linearity of the multidimensional Riemann integral: Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ .

Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, §2.4.2, pp. 38–39: full lecture notes, PDF.

Darryl D. Holm, Ideal shallow water dynamics in a rotating frame, §§1.1 and 1.4, pp. 3–6: full lecture notes, PDF.

Paola Rizzoli, MIT OpenCourseWare 12.802, Lecture 12, pp. 1–4: full lecture notes, PDF.

Facts & Assumptions

Given: The Axiom of Countable Choice ${A C}_{ω}$ ; $h > 0$ and $h, v$ are classical and $2 π$ -periodic in space; $f \in R$ and $g > 0$ are constant; the flat-bottom, inviscid equations in the statement hold pointwise.

[F1]

Rotating stratified and shallow-water equation families: the shallow-water branch is $h_{t} + \nabla \cdot (h v) = 0$ and $(h v)_{t} + \nabla \cdot (h v \otimes v) + \nabla (g h^{2} / 2) + f h J v = 0$ , with $h > 0$ , $g > 0$ , and $f \in R$ constant.

[F2]

Rotating stratified and shallow-water equation families: for scalar $q$ , $\nabla q = (\partial_{x_{1}} q, \partial_{x_{2}} q)$ ; for vector $w$ , $\nabla \cdot w = \sum_{j} \partial_{x_{j}} w_{j}$ ; for a matrix $A$ , $(\nabla \cdot A)_{i} = \sum_{j} \partial_{x_{j}} A_{i j}$ ; and $J (v_{1}, v_{2}) = (- v_{2}, v_{1})$ .

[A1]

The fields have continuous derivatives through the orders appearing below; division by $h$ is valid because $h > 0$ .

[A2]

Countable Choice ${A C}_{ω}$ means that every $N$ -indexed family of nonempty sets has a choice function; it is used for the cited Lebesgue-measure and Riemann-to-Lebesgue comparison results (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

Every continuous real function on a closed nondegenerate rectangle in $R^{m}$ , $m \geq 1$ , is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable).

[L2]

Assuming ${A C}_{ω}$ , every bounded Borel Riemann-integrable real function on a closed nondegenerate box is Lebesgue integrable and its Lebesgue and Riemann integrals agree (Borel Darboux integrands in finite dimension).

[L3]

For a Riemann-integrable function on a product of nondegenerate closed rectangles, if every section in a chosen order is Riemann integrable, the ordinary iterated Riemann integral in that order exists and equals the product Riemann integral; integrability on the product alone does not guarantee that every section is integrable (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[L4]

If $G$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $G^{'}$ has a Riemann-integrable extension $f$ to $[a, b]$ , then $\int_{a}^{b} f = G (b) - G (a)$ (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative).

[L5]

The multidimensional Riemann integral is linear on integrable functions over a nondegenerate rectangle (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

Proof

1.1givenF1F2A1algebra

Write $D_{t} : = \partial_{t} + v \cdot \nabla$ . By the product rule and the component conventions in [F2], the mass equation is $D_{t} h = - h \nabla \cdot v$ , while $(h v)_{t} + \nabla \cdot (h v \otimes v) = h D_{t} v + v (h_{t} + \nabla \cdot (h v)) = h D_{t} v$ . Thus the momentum equation and $\nabla (g h^{2} / 2) = g h \nabla h$ give $D_{t} v = - f J v - g \nabla h$ .

2.1givenF1F2A1step 1.1algebra

Taking the dot product with $v$ and using $v \cdot J v = 0$ yields $D_{t} (∣ v ∣^{2} / 2) = - g v \cdot \nabla h$ . The product-rule identity $\partial_{t} (h q) + \nabla \cdot (h q v) = h D_{t} q + q (h_{t} + \nabla \cdot (h v))$ and continuity with $q = ∣ v ∣^{2} / 2$ give $\partial_{t} (h ∣ v ∣^{2} / 2) + \nabla \cdot ((h ∣ v ∣^{2} / 2) v) = - g h v \cdot \nabla h$ . Similarly, applying $D_{t}$ to $g h^{2} / 2$ and using $D_{t} h = - h \nabla \cdot v$ gives $\partial_{t} (g h^{2} / 2) + \nabla \cdot ((g h^{2} / 2) v) = - (g h^{2} / 2) \nabla \cdot v$ . Adding these balances and using $\nabla \cdot ((g h^{2} / 2) v) = g h v \cdot \nabla h + (g h^{2} / 2) \nabla \cdot v$ proves the stated local flux identity.

3.1givenA1A2F1F2L1L2L3L4L5step 2.1algebra∎

Set $K = [t_{0}, t_{1}] \times Q_{2}$ and write $R_{A} (F)$ for the Riemann integral of a Riemann-integrable $F$ over a rectangle $A$ . The local flux identity from step 2.1 has continuous scalar terms on $K$ , which are Riemann integrable by [L1]. Integrate that identity over $K$ as a Riemann integral. For each fixed $x$ , [L4] in time gives $\int_{t_{0}}^{t_{1}} \partial_{t} E (t, x) d t = E (t_{1}, x) - E (t_{0}, x)$ ; [L3] then gives $R_{K} (\partial_{t} E) = R_{Q_{2}} (E (t_{1}, \cdot) - E (t_{0}, \cdot))$ . For each spatial flux component and each fixed value of the other coordinates, [L4] in the differentiated coordinate gives the endpoint difference at $0$ and $2 π$ , which is zero by periodicity; [L3] yields $R_{K} (\nabla \cdot ((E + g h^{2} / 2) v)) = 0$ . By [L5], the integrated local identity therefore gives $R_{Q_{2}} (E (t_{1}, \cdot)) = R_{Q_{2}} (E (t_{0}, \cdot))$ . Each endpoint energy density is continuous and hence Borel on $Q_{2}$ ; [L2] under [A2] identifies these Riemann integrals with the Lebesgue integrals in the statement.

Smooth rotating shallow-water energy balance

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources