A one-dimensional C^3 example shows that vanishing-viscosity limits for mechanical Hamiltonian ergodic problems may not exist.
Bolotin, New Connections Between Dynamical Systems and PDEs, American Institute of Mathematics Workshop Open Problem List,https://aimath.org/WWN/dynpde/dynpde.pdf
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Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems
A one-dimensional C^3 example shows that vanishing-viscosity limits for mechanical Hamiltonian ergodic problems may not exist.