Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

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

Research output: Contribution to journalArticlepeer-review

2 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 languageEnglish (US)
Pages (from-to)71-111
Number of pages41
JournalArchive for Rational Mechanics and Analysis
Volume217
Issue number1
DOIs
StatePublished - Dec 27 2014

ASJC Scopus subject areas

  • Mechanical Engineering
  • Analysis
  • Mathematics (miscellaneous)

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