Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Claude W. Bardos; François Golse; Peter A. Markowich; Thierry A. Paul

doi:10.1007/s00205-014-0829-7

Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Claude W. Bardos, François Golse, Peter A. Markowich, Thierry A. Paul

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4 Scopus citations

Abstract

Consider a monokinetic probability measure on the phase space (Formula presented.) , i.e. (Formula presented.) where Uin is a vector field on RN and ρin a probability density on RN. Let Φt be a Hamiltonian flow on RN × RN. In this paper, we study the structure of the transported measure (Formula presented.) and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in (Formula presented.) , on which μ(t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles Uin with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp.

Original language	English (US)
Pages (from-to)	71-111
Number of pages	41
Journal	Archive for Rational Mechanics and Analysis
Volume	217
Issue number	1
DOIs	https://doi.org/10.1007/s00205-014-0829-7
State	Published - Dec 27 2014

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Peter Markowich thanks the Fondation des Sciences Mathematiques de Paris for its support during the preparation of this paper.

ASJC Scopus subject areas

Mechanical Engineering
Analysis
Mathematics (miscellaneous)

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title = "Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile",

abstract = "Consider a monokinetic probability measure on the phase space (Formula presented.) , i.e. (Formula presented.) where Uin is a vector field on RN and ρin a probability density on RN. Let Φt be a Hamiltonian flow on RN × RN. In this paper, we study the structure of the transported measure (Formula presented.) and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in (Formula presented.) , on which μ(t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schr{\"o}dinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles Uin with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp.",

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AU - Bardos, Claude W.

AU - Golse, François

AU - Markowich, Peter A.

AU - Paul, Thierry A.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Peter Markowich thanks the Fondation des Sciences Mathematiques de Paris for its support during the preparation of this paper.

PY - 2014/12/27

Y1 - 2014/12/27

N2 - Consider a monokinetic probability measure on the phase space (Formula presented.) , i.e. (Formula presented.) where Uin is a vector field on RN and ρin a probability density on RN. Let Φt be a Hamiltonian flow on RN × RN. In this paper, we study the structure of the transported measure (Formula presented.) and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in (Formula presented.) , on which μ(t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles Uin with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp.

AB - Consider a monokinetic probability measure on the phase space (Formula presented.) , i.e. (Formula presented.) where Uin is a vector field on RN and ρin a probability density on RN. Let Φt be a Hamiltonian flow on RN × RN. In this paper, we study the structure of the transported measure (Formula presented.) and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in (Formula presented.) , on which μ(t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles Uin with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp.

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Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Abstract

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