An Optimal Transport Approach for the Kinetic Bohmian Equation
- 3 Downloads
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.
Unable to display preview. Download preview PDF.
- 5.P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, and A. Porretta, “Long time average of mean field games with a nonlocal coupling,” Control and Optimization, SIAM, 51, 3558–3591 (2013).Google Scholar
- 6.T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes, 10, Amer. Math. Soc. (2003).Google Scholar
- 7.J. Cushing, S. Goldstein, and A. Fine (eds.), Bohmian Mechanics and Quantum Theory: an Appraisal, Springer Science Business Media, 184 (2013).Google Scholar
- 8.D. Dèurr and S. Teufel, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer (2009).Google Scholar
- 14.E. Lieb, R. Seiringer, and J. Yngvason, “Bosons in a trap: A rigorous derivation of the Gross–Pitaevskii energy functional,” in: The Stability of Matter: From Atoms to Stars, Springer (2001), pp. 685–697.Google Scholar
- 18.F. Otto, The Geometry of Dissipative Evolution Equations: the Porous Medium Equation, Taylor and Francis (2001).Google Scholar
- 20.C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, 139, Springer, New York (1999).Google Scholar
- 21.T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, Amer. Math. Soc. (2006).Google Scholar
- 23.C. Villani, Topics in Optimal Transportation, Graduate Studies in Math., 58, Amer. Math. Soc. (2003).Google Scholar