On the Quantum Boltzmann Equation near Maxwellian and Vacuum

TL;DR

This paper establishes the global well-posedness of the non-relativistic quantum Boltzmann equation for fermions and bosons near Maxwellian and vacuum, extending classical Boltzmann theory to quantum kinetic equations.

Contribution

It provides a detailed analysis of the nonlinear operator in the quantum setting and justifies well-posedness near Maxwellian and vacuum states.

Findings

Proves global existence and uniqueness of solutions.

Analyzes the nonlinear operator Q in quantum context.

Extends classical Boltzmann theory to quantum kinetic equations.

Abstract

We consider the non-relativistic quantum Boltzmann equation for fermions and bosons. Using the nonlinear energy method and mild formulation, we justify the global well-posedness when the density function is near the global Maxwellian and vacuum. This work is a generalization and adaptation of the classical Boltzmann theory. Our main contribution is a detailed analysis of the nonlinear operator $Q$ in the quantum context. This is the first piece of a long-term project on the quantum kinetic equations.