Discoveries / differential-equations

The Jauslin-Kreiss-Moser Vanishing-Viscosity Selection Problem

For the ergodic problem $\tfrac12|D\varphi^\varepsilon|^2 + F(x) - \varepsilon\Delta\varphi^\varepsilon = c(\varepsilon)$ on the torus, normalized by $\varphi^\varepsilon(0) = 0$, Jauslin, Kreiss and Moser asked whether the vanishing-viscosity limit $\lim_{\varepsilon \to 0}\varphi^\varepsilon$ always exists. It need not: there is a one-dimensional example with $F \in C^3$ for which the limit fails to exist.