Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile
Type
ArticleKAUST Department
Applied Mathematics and Computational Science ProgramComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Date
2014-12-27Online Publication Date
2014-12-27Print Publication Date
2015-07Permanent link to this record
http://hdl.handle.net/10754/566116
Metadata
Show full item recordAbstract
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 NaturearXiv
arXiv:1207.5927v2ae974a485f413a2113503eed53cd6c53
10.1007/s00205-014-0829-7