Finite-time blow-up and variational approximation scheme for a Wigner-Fokker-Planck equation with a nonlocal perturbation

The paper is concerned with analysis of a Wigner-Fokker-Planck equation with a nonlocal perturbation that consists both conservative and dissipative as well as nonlinear non-local effects. We first show that the system has a finite-time blow-up phenomenon. We then introduce a variational steepest descent approximation scheme for the system. At each step, the scheme minimizes an energy functional with respect to the Kantorovich functional associated with a certain cost function which is inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system.