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Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Abstract : Consider 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. Finally, we discuss applications of our results to the classical limit of the Schrödinger equation.
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https://hal-polytechnique.archives-ouvertes.fr/hal-00706180
Contributor : François Golse Connect in order to contact the contributor
Submitted on : Wednesday, July 25, 2012 - 10:43:43 AM
Last modification on : Monday, October 26, 2020 - 2:50:03 PM
Long-term archiving on: : Friday, October 26, 2012 - 2:21:04 AM

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  • HAL Id : hal-00706180, version 2

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Claude Bardos, François Golse, Peter Markowich, Thierry Paul. Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile. 2012. ⟨hal-00706180v2⟩

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