caa a22 4500 605515492 CHVBK 20210128100709.0 cr unu---uuuuu 210128e20150701xx s 000 0 eng 10.1007/s00205-014-0829-7 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0829-7 Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile [Elektronische Daten] [Claude Bardos, François Golse, Peter Markowich, Thierry Paul] Consider a monokinetic probability measure on the phase space $${{\bf R}^N_{x} \times {\bf R}^N_{\xi}}$$ R x N × R ξ N , i.e. $${\mu^{\rm {in}} = \rho^{\rm {in}}(x)\delta(\xi - U^{\rm {in}}(x))}$$ μ in = ρ in ( x ) δ ( ξ - U in ( x ) ) where U in is a vector field on R N and ρ in a probability density on R N . Let Φ t be a Hamiltonian flow on R N × R N . In this paper, we study the structure of the transported measure $${\mu(t) := \Phi_t\#\mu^{\rm {in}}}$$ μ ( t ) : = Φ t # μ in and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in $${\Phi_t({\rm graph of} U^{\rm {in}})}$$ Φ t ( graph of U in ) , on whichμ ( 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 U in with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp. Springer-Verlag Berlin Heidelberg, 2014 Bardos Claude Laboratoire J.-L. Lions, Université Paris-Diderot, 4 place Jussieu, BP187, 75252, Paris Cedex 05, France aut Golse François Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz (CMLS), 91128, Palaiseau Cedex, France aut Markowich Peter MCSE Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Saudi Arabia aut Paul Thierry CNRS and Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz (CMLS), 91128, Palaiseau Cedex, France aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/1(2015-07-01), 71-111 0003-9527 217:1<71 2015 217 205 https://doi.org/10.1007/s00205-014-0829-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0829-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bardos Claude Laboratoire J.-L. Lions, Université Paris-Diderot, 4 place Jussieu, BP187, 75252, Paris Cedex 05, France aut NATIONALLICENCE 700 1- Golse François Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz (CMLS), 91128, Palaiseau Cedex, France aut NATIONALLICENCE 700 1- Markowich Peter MCSE Division, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Saudi Arabia aut NATIONALLICENCE 700 1- Paul Thierry CNRS and Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz (CMLS), 91128, Palaiseau Cedex, France aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/1(2015-07-01), 71-111 0003-9527 217:1<71 2015 217 205