### Abstract

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.

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Technische Universiteit Eindhoven |

Number of pages | 19 |

Publication status | Published - 2014 |

### Publication series

Name | CASA-report |
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Volume | 1434 |

ISSN (Print) | 0926-4507 |

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## Cite this

Lith, van, B. S., Thije Boonkkamp, ten, J. H. M., IJzerman, W. L., & Tukker, T. W. (2014).

*Existence and uniqueness of solutions to Liouville's equation and the associated flow for Hamiltonians of bounded variation*. (CASA-report; Vol. 1434). Technische Universiteit Eindhoven.