https://hal-polytechnique.archives-ouvertes.fr/hal-00706180v3Bardos, ClaudeClaudeBardosLJLL - Laboratoire Jacques-Louis Lions - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueGolse, FrançoisFrançoisGolseCMLS - Centre de Mathématiques Laurent Schwartz - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueMarkowich, PeterPeterMarkowichMCSE Division - KAUST - King Abdullah University of Science and TechnologyPaul, ThierryThierryPaulCMLS - Centre de Mathématiques Laurent Schwartz - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueHamiltonian Evolution of Monokinetic Measures with Rough Momentum ProfileHAL CCSD2015Wigner measureLiouville equationSchrödinger equationWKB methodCausticArea formulaCoarea formula[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP][MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]Golse, François2013-04-26 14:18:172023-02-07 14:44:492013-04-26 15:01:57enJournal articleshttps://hal-polytechnique.archives-ouvertes.fr/hal-00706180v3/document10.1007/s00205-014-0829-7https://hal-polytechnique.archives-ouvertes.fr/hal-00706180v2application/pdf3Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.