TY - JOUR

T1 - An Optimal Transport Approach for the Kinetic Bohmian Equation

AU - Gangbo, W.

AU - Haskovec, Jan

AU - Markowich, Peter A.

AU - Sierra Nunez, Jesus Alfredo

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The research of W. Gangbo was supported by the NSF grant DMS–1160939.

PY - 2019/3/23

Y1 - 2019/3/23

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/652828

UR - http://link.springer.com/article/10.1007/s10958-019-04248-3

UR - http://www.scopus.com/inward/record.url?scp=85065287044&partnerID=8YFLogxK

U2 - 10.1007/s10958-019-04248-3

DO - 10.1007/s10958-019-04248-3

M3 - Article

SN - 1072-3374

VL - 238

SP - 415

EP - 452

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

IS - 4

ER -