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Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile
- Source :
- Archive for Rational Mechanics and Analysis, Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 217 (1), pp.71-111. ⟨10.1007/s00205-014-0829-7⟩, Archive for Rational Mechanics and Analysis, 2015, 217 (1), pp.71-111. ⟨10.1007/s00205-014-0829-7⟩
- Publication Year :
- 2012
-
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 Schr\"{o}dinger equation. Finally we present various examples and counterexamples showing that our results are sharp.<br />Comment: 35 pages; main theorems gathered in section 2; examples and counterexamples gathered in section 3; examples 3.1 and 3.4 added; example 3.3 extended to the case of smooth momentum profiles; proof of Maslov's Theorem 1.1 (formerly Proposition 7.4) removed; some typos corrected
- Subjects :
- WKB method
FOS: Physical sciences
Schrödinger equation
Caustic
Probability density function
Area formula
01 natural sciences
Classical limit
symbols.namesake
Mathematics (miscellaneous)
Mathematics - Analysis of PDEs
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
FOS: Mathematics
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
0101 mathematics
Mathematical Physics
Mathematics
Lebesgue measure
MSC 81Q20, 81S30, 35Q40, 35L03, 28A75
Mechanical Engineering
010102 general mathematics
Mathematical analysis
Coarea formula
Mathematical Physics (math-ph)
16. Peace & justice
Lipschitz continuity
010101 applied mathematics
Liouville equation
Phase space
Radon measure
symbols
81Q20, 81S30, 35Q40, 35L03, 28A75
Configuration space
Hamiltonian (quantum mechanics)
Wigner measure
Analysis
Analysis of PDEs (math.AP)
Subjects
Details
- Language :
- English
- ISSN :
- 00039527 and 14320673
- Database :
- OpenAIRE
- Journal :
- Archive for Rational Mechanics and Analysis, Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 217 (1), pp.71-111. ⟨10.1007/s00205-014-0829-7⟩, Archive for Rational Mechanics and Analysis, 2015, 217 (1), pp.71-111. ⟨10.1007/s00205-014-0829-7⟩
- Accession number :
- edsair.doi.dedup.....242ad44e33c40a32ab763cf3cd4ae1ea