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    Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

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    Type
    Article
    Authors
    Bardos, Claude W.
    Golse, François
    Markowich, Peter A. cc
    Paul, Thierry A.
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2014-12-27
    Online Publication Date
    2014-12-27
    Print Publication Date
    2015-07
    Permanent link to this record
    http://hdl.handle.net/10754/566116
    
    Metadata
    Show full item record
    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.
    Sponsors
    Peter Markowich thanks the Fondation des Sciences Mathematiques de Paris for its support during the preparation of this paper.
    Publisher
    Springer Nature
    Journal
    Archive for Rational Mechanics and Analysis
    DOI
    10.1007/s00205-014-0829-7
    arXiv
    arXiv:1207.5927v2
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00205-014-0829-7
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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