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arXiv · 2605.10478

Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems

Ziran Liu·Hung V. Tran·Yifeng Yu
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Abstract

For $\varepsilon>0$, let $\phi^\varepsilon$ be the solution of the ergodic problem \[ \frac12 |D\phi^\varepsilon|^2+F(x)-\varepsilon\Delta\phi^\varepsilon=c(\varepsilon) \qquad \text{on } \mathbb{T}^n, \] normalized by $\phi^\varepsilon(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit $\lim_{\varepsilon\to0}\phi^\varepsilon$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10].

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Ziran Liu, Hung V. Tran, Yifeng Yu. 2026-05-11. Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems. https://arxiv.org/abs/2605.10478

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