# Stationary quantum BGK model for bosons and fermions in a bounded interval

## TL;DR

This paper proves the existence of solutions for a stationary quantum BGK model describing bosons and fermions in a bounded interval, ensuring the quantum states are well-defined without phase transitions.

## Contribution

It establishes the existence of mild solutions for the quantum BGK model with boundary data, addressing quantum-specific equilibrium states and phase transition issues.

## Key findings

- Existence of mild solutions for quantum BGK models
- Well-defined quantum local equilibrium states
- No phase transition issues in the solution space

## Abstract

In this paper, we consider the existence problem for a stationary relaxational models of the quantum Boltzmann equation. More precisely, we establish the existence of mild solution to the fermionic or bosonic quantum BGK model in a slab with inflow boundary data. Unlike the classical case, it is necessary to verify that the quantum local equilibrium state is well-defined, and the transition from the non-condensed state to the condensated state (Bosons), or from the non-saturated state to the saturated state (Fermions) does not arise in our solution space.