Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems

Abstract

For $\epsilon >0$, let $\ddot{I}{•}^{\epsilon }$ 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 $\ddot{I}{•}^{\epsilon }(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit ${\lim }_{\epsilon \to 0}\ddot{I}{•}^{\epsilon }$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10].

15 pages. The authors used ChatGPT during the development of this work, including for suggesting proof strategies and assisting with calculations. All mathematical statements, proofs, and verifications were subsequently completed and rigorously checked by the authors, who take full responsibility for the results