Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems

TL;DR

This paper constructs a one-dimensional example demonstrating that the vanishing-viscosity limit does not always exist for certain ergodic Hamiltonian problems, challenging previous assumptions.

Contribution

It provides a specific counterexample showing the nonexistence of the vanishing-viscosity limit in ergodic Hamiltonian problems, answering an open question.

Findings

A constructed example where the limit of solutions as viscosity vanishes does not exist.

The example involves a smooth potential function F in a one-dimensional setting.

This result disproves a previously posed conjecture about the limits in such problems.

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].