Well-posedness for the spatially inhomogeneous Fermi-Dirac-Fokker-Planck equation

This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor $f(1-f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint $0\le f\le 1$. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation $(BV),$ allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.