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 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.

Citation
Bardos, C., Golse, F., Markowich, P., & Paul, T. (2014). Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile. Archive for Rational Mechanics and Analysis, 217(1), 71–111. doi:10.1007/s00205-014-0829-7

Acknowledgements
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
1207.5927

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