Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Claude Bardos 1 François Golse 2 Peter Markowich 3 Thierry Paul 2
1 LJLL - Laboratoire Jacques-Louis Lions
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 MCSE Division
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. 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.
Keywords : Wigner measure Liouville equation Schrödinger equation WKB method Caustic Area formula Coarea formula
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Journal articles
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https://hal-polytechnique.archives-ouvertes.fr/hal-00706180
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Submitted on : Friday, April 26, 2013 - 2:18:17 PM
Last modification on : Sunday, March 31, 2019 - 1:36:35 AM
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Claude Bardos, François Golse, Peter Markowich, Thierry Paul. Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile. Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 217 (1), pp.71-111. ⟨10.1007/s00205-014-0829-7⟩. ⟨hal-00706180v3⟩

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