Quantum BGK model near a global Fermi-Dirac distribution

TL;DR

This paper studies the fermionic quantum BGK model, proving existence and exponential stabilization of solutions near a global Fermi-Dirac distribution, addressing unique equilibrium parameter determination.

Contribution

It establishes the existence and exponential decay of solutions to the quantum BGK model near equilibrium, handling nonlinear equilibrium parameter determination.

Findings

Existence of unique classical solutions near equilibrium.

Exponential stabilization of solutions.

Resolution of nonlinear equilibrium parameter determination.

Abstract

In this paper, we consider the existence and asymptotic behavior of the fermionic quantum BGK model, which is a relaxation model of the quantum Boltzmann equation for fermions. More precisely, we establish the existence of unique classical solutions and their exponentially fast stabilization when the initial data starts sufficiently close to a global Fermi-Dirac distribution. A key difficulty unobserved in the study of the classical BGK model is that we must verify that the equilibrium parameters is uniquely determined through a set of nonlinear equations in each iteration step.