Existence and uniqueness of solutions to Liouville's equation and the associated flow for Hamiltonians of bounded variation

摘要: We prove existence and uniqueness for solutions to Liouville's equation for Hamiltonians of bounded variation. These solutions can be interpreted as the limit of a sequence generated by a series of smooth approximations to the Hamiltonian. This results in a converging sequence of approximations of solutions to Liouville's equation. As an added perk, our method allows us to prove a generalisation of Liouville's theorem for Hamiltonians of bounded variation. Furthermore, we prove there exists a unique flow solution to the Hamilton equations and show how this can be used to construct a solution to Liouville's equation. Key words: partial differential equations, geometrical optics, Liouville's equation, flow.