An Optimal Transport Approach for the Kinetic Bohmian Equation

Gangbo, W.; Haskovec, Jan; Markowich, Peter A.; Sierra Nunez, Jesus Alfredo

An Optimal Transport Approach for the Kinetic Bohmian Equation

Files

p114-SudakovMemorial.pdf (422.3 KB)

Type

Article

Authors

Gangbo, W.
Haskovec, Jan
Markowich, Peter A.

Sierra Nunez, Jesus Alfredo

KAUST Department

Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program

Online Publication Date

2019-03-23

Print Publication Date

2019-04

Date

2019-03-23

Abstract

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.

Citation

Gangbo W, Haskovec J, Markowich P, Sierra J (2019) An Optimal Transport Approach for the Kinetic Bohmian Equation. Journal of Mathematical Sciences 238: 415–452. Available: http://dx.doi.org/10.1007/s10958-019-04248-3.

Acknowledgements

The research of W. Gangbo was supported by the NSF grant DMS–1160939.