Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems

For $\epsilon >0$, let ${\phi }^{\epsilon }$ be the solution of the ergodic problem $$\frac{1}{2}\mathrm{\mid }D{\phi }^{\epsilon }{\mathrm{\mid }}^2+F(x)-\epsilon \mathrm{\Delta }{\phi }^{\epsilon }=c(\epsilon )\text{on }{\mathbb{T}}^n,$$ normalized by ${\phi }^{\epsilon }(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit ${\lim }_{\epsilon \to 0}{\phi }^{\epsilon }$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10].