On a nonlinear parabolic problem arising in the quantum diffusive description of a degenerate fermion gas

TL;DR

This paper investigates a nonlinear parabolic equation modeling a degenerate fermion gas, analyzing existence of stationary solutions and long-term behavior through theoretical proofs and numerical simulations.

Contribution

It provides the first combined theoretical and numerical analysis of a nonlinear drift-diffusion equation in quantum fermion gas modeling.

Findings

Existence of stationary solutions under certain boundary conditions.

Numerical evidence of long-time behavior and convergence.

Insights into quantum diffusive processes in fermion gases.

Abstract

This article studies, both theoretically and numerically, a nonlinear drift-diffusion equation describing a gas of fermions in the zero-temperature limit. The equation is considered on a bounded domain whose boundary is divided into an "insulating" part, where homogeneous Neumann conditions are imposed, and a "contact" part, where nonhomogeneous Dirichlet data are assigned. The existence of stationary solutions for a suitable class of Dirichlet data is proven by assuming a simple domain configuration. The long-time behavior of the time-dependent solution, for more complex domain configurations, is investigated by means of numerical experiments.