The title is self explaining: I see in a lot of literature that PDEs with some Hamiltonian structure in it are solved over a torus or some other compact manifold. Why is that? At least I now that it has something to do with the case where the Hamiltonian fails to be coercive. I have pretty good notions of the HJB equation in $\mathbb R^n.$
I want to immerse myself in this theory, so the question aims more to get some advice about good literature.