An Optimal Transport Approach for the Kinetic Bohmian Equation
- Authors: Gangbo W.1, Haskovec J.2, Markowich P.2, Sierra J.2
-
Affiliations:
- University of California at Los Angeles
- CEMSE Division, King Abdullah University of Science and Technology
- Issue: Vol 238, No 4 (2019)
- Pages: 415-452
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/242541
- DOI: https://doi.org/10.1007/s10958-019-04248-3
- ID: 242541
Cite item
Abstract
We study the existence theory of solutions of the kinetic Bohmian equation, a nonlinear Vlasov-type equation proposed for the phase-space formulation of Bohmian mechanics. Our main idea is to interpret the kinetic Bohmian equation as a Hamiltonian system defined on an appropriate Poisson manifold built on a Wasserstein space. We start by presenting an existence theory for stationary solutions of the kinetic Bohmian equation. Afterwards, we develop an approximative version of our Hamiltonian system in order to study its associated flow. We then prove the existence of solutions of our approximative version. Finally, we present some convergence results for the approximative system; our aim is to show that, in the limit, the approximative solution satisfies the kinetic Bohmian equation in a weak sense.
About the authors
W. Gangbo
University of California at Los Angeles
Author for correspondence.
Email: wgangbo@math.ucla.edu
United States, Los Angeles
J. Haskovec
CEMSE Division, King Abdullah University of Science and Technology
Email: wgangbo@math.ucla.edu
Saudi Arabia, Thuwal
P. Markowich
CEMSE Division, King Abdullah University of Science and Technology
Email: wgangbo@math.ucla.edu
Saudi Arabia, Thuwal
J. Sierra
CEMSE Division, King Abdullah University of Science and Technology
Email: wgangbo@math.ucla.edu
Saudi Arabia, Thuwal