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| HAL: hal-00706180, version 2 |
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| Available versions: | v1 (2012-06-11) | v2 (2012-07-25) | v3 (2013-04-26) |
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| Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile |
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| Claude Bardos 1François Golse 1, 2 |
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| (2012-06-09) |
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| 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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| 1: | Laboratoire Jacques-Louis Lions (LJLL) |
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CNRS : UMR7598 – Université Pierre et Marie Curie [UPMC] - Paris VI |
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| 2: | Centre de Mathématiques Laurent Schwartz (CMLS-EcolePolytechnique) |
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CNRS : UMR7640 – Polytechnique - X |
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| 3: | Department of Applied Mathematics and Theoretical Physics (DAMTP) |
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University of Cambridge |
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| 4: | MCSE Division |
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King Abdullah University of Science and Technology (KAUST) |
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| Subject | : | Mathematics/Analysis of PDEs Mathematics/Mathematical Physics |
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| Wigner measure – Liouville equation – Schrödinger equation – WKB method – Caustic – Area formula – Coarea formula |
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| hal-00706180, version 2 | |
| http://hal-polytechnique.archives-ouvertes.fr/hal-00706180 | |
| oai:hal-polytechnique.archives-ouvertes.fr:hal-00706180 | |
| From: François Golse | |
| Submitted on: Wednesday, 25 July 2012 10:43:43 | |
| Updated on: Wednesday, 25 July 2012 11:39:55 | |
