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The Jauslin-Kreiss-Moser Vanishing-Viscosity Selection Problem

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jauslin-kreiss-moser-vanishing-viscosityDifferential equationsposed by Hans R. Jauslin, Heinz-Otto Kreiss, Jurgen Moser, 1999recorded: disproved

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Statement

For the ergodic problem 12∣Dφε∣2+F(x)−εΔφε=c(ε)\tfrac12|D\varphi^\varepsilon|^2 + F(x) - \varepsilon\Delta\varphi^\varepsilon = c(\varepsilon) on the torus, normalized by φε(0)=0\varphi^\varepsilon(0) = 0, Jauslin, Kreiss and Moser asked whether the vanishing-viscosity limit lim⁡ε→0φε\lim_{\varepsilon \to 0}\varphi^\varepsilon always exists. It need not: there is a one-dimensional example with F∈C3F \in C^3 for which the limit fails to exist.

Context

A selection question for viscous Hamilton-Jacobi equations posed in a 1999 Proceedings of Symposia in Pure Mathematics article and open since; vanishing-viscosity selection is the standard route to picking a weak KAM solution.

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  • #1

    Attempt 1

    constructionChatGPT with Ziran Liu, Hung V. Tran, Yifeng Yu ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    ChatGPT
    people
    Ziran Liu, Hung V. Tran, Yifeng Yu

    Generic rather than itemised, and prominent enough that the authors put it in the arXiv comment as well as the acknowledgement: they used ChatGPT during the development of the work, including for suggesting proof strategies and assisting with calculations, and then completed and rigorously checked every statement, proof and verification themselves. No model version is named and no individual step is attributed, which is what keeps this at the assistive end rather than higher.

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