Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913112566792192 |
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| author | Liu, Ziran Tran, Hung V. Yu, Yifeng |
| author_facet | Liu, Ziran Tran, Hung V. Yu, Yifeng |
| contents | For $\varepsilon>0$, let $ϕ^\varepsilon$ be the solution of the ergodic problem \[
\frac12 |Dϕ^\varepsilon|^2+F(x)-\varepsilonΔϕ^\varepsilon=c(\varepsilon)
\qquad \text{on } \mathbb{T}^n, \] normalized by $ϕ^\varepsilon(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit $\lim_{\varepsilon\to0}ϕ^\varepsilon$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10478 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems Liu, Ziran Tran, Hung V. Yu, Yifeng Analysis of PDEs For $\varepsilon>0$, let $ϕ^\varepsilon$ be the solution of the ergodic problem \[ \frac12 |Dϕ^\varepsilon|^2+F(x)-\varepsilonΔϕ^\varepsilon=c(\varepsilon) \qquad \text{on } \mathbb{T}^n, \] normalized by $ϕ^\varepsilon(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit $\lim_{\varepsilon\to0}ϕ^\varepsilon$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10]. |
| title | Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.10478 |