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.

A strong local L1 limit preserves shallow-water dry admissibility

Statement

Let $h_{k} \geq 0$ almost everywhere on $R \times (0, T)$ and suppose $h_{k} \to h$ strongly in $L_{l o c}^{1}$ . Let $u$ be measurable, put $m = h u$ almost everywhere, and suppose $u = 0$ almost everywhere on ${h = 0}$ . Then $U = (h, m)$ belongs to the admissible state set $K$ from One-dimensional shallow-water balance law with a dry-state convention almost everywhere. Also, let initial reconstructions $U_{k}^{0} = (h_{k}^{0}, m_{k}^{0}) \in K$ satisfy $m_{k}^{0} = h_{k}^{0} u_{k}^{0}$ , $u_{k}^{0} = 0$ where $h_{k}^{0} = 0$ , and $∣ u_{k}^{0} ∣ \leq M_{Q}$ on each compact spatial interval $Q$ for a constant $M_{Q}$ independent of $k$ . If $U_{k}^{0} \to U_{0} = (h_{0}, m_{0})$ strongly in $L_{l o c}^{1} (R)$ , then $U_{0} \in K$ almost everywhere.

Facts & Assumptions

Given: Nonnegative bulk depths converging strongly locally in $L^{1}$ , $m = h u$ with zero velocity on the dry set, and admissible initial reconstructions converging strongly locally in $L^{1}$ with locally uniform primitive-velocity bounds. Prove both limit states are admissible.

[F1]

One-dimensional shallow-water balance law with a dry-state convention: $K = {(h, m) : h > 0, m \in R} \cup {(0, 0)}$ , so any positive-depth state is admissible and zero depth requires zero momentum.

Proof

1.1givenalgebra

Write $r_{-} : = \max (- r, 0)$ . On each compact rectangle $Q \subset R \times (0, T)$ , $h_{k} \geq 0$ implies $h_{-} \leq ∣ h - h_{k} ∣$ , so $\int_{Q} h_{-} \leq ∥ h - h_{k} ∥_{L^{1} (Q)} \to 0$ . On each compact interval $I \subset R$ , $h_{k}^{0} \geq 0$ gives $(h_{0})_{-} \leq ∣ h_{0} - h_{k}^{0} ∣$ , so $\int_{I} (h_{0})_{-} \leq ∥ h_{0} - h_{k}^{0} ∥_{L^{1} (I)} \to 0$ . Countable compact exhaustions of $R \times (0, T)$ and $R$ now give $h \geq 0$ and $h_{0} \geq 0$ almost everywhere on their domains.

2.1F1step 1.1givenalgebra∎

Where $h > 0$ , [F1] allows arbitrary momentum $m$ ; where $h = 0$ , $m = h u = 0$ , so $U \in K$ . For the initial data, on each compact interval $I$ the bound on $u_{k}^{0}$ and $U_{k}^{0} \in K$ imply $∣ m_{k}^{0} ∣ \leq M_{I} h_{k}^{0}$ . Hence $(∣ m_{0} ∣ - M_{I} h_{0})_{+} \leq ∣ m_{0} - m_{k}^{0} ∣ + M_{I} ∣ h_{k}^{0} - h_{0} ∣$ pointwise, and integration followed by strong $L^{1} (I)$ convergence gives $∣ m_{0} ∣ \leq M_{I} h_{0}$ almost everywhere. Thus $m_{0} = 0$ wherever $h_{0} = 0$ ; step 1.1 gives $h_{0} \geq 0$ , and [F1] yields $U_{0} \in K$ . A countable compact exhaustion proves the bulk and initial claims almost everywhere on their domains.

Depends on

One-dimensional shallow-water balance law with a dry-state convention

Used by

A strong limit of hydrostatic HLL updates solves the smooth-bed balance law weakly Theorem

Dependency tree · one level

1 result within one dependency step 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

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows (standard reference, not scraped)
D. I. Ketcheson, R. J. LeVeque, and M. J. del Razo, Riemann Problems and Jupyter Solutions, Chapter 12 (standard reference, not scraped)