An optimal transport approach for the kinetic Bohmian equation (Q2313793)

The paper addresses the problem of the existence of solutions to a kinetic equation which originates from an alternative (Bohmian) formulation of quantum mechanics and has a form similar to the Vlasov's equations, which are well known in plasma physics. These are nonlinear partial differential equations for the distribution function of coordinates and components of the velocity vector in the appropriate phase space. The equations include derivatives with respect to both the coordinates and velocities. The objective of the consideration is to rigorously prove the existence of solutions to this equation with appropriate initial conditions. First, the equation is approximated by a Hamiltonian system, for which the existence of the solution can be proved. Then, it is proved that the approximate solution converges to an exact one, following the transition from the approximate system to the exact underlying kinetic equation.