An explicit coercivity estimate of the linearized quantum Boltzmann operator without angular cutoff

TL;DR

This paper establishes an explicit coercivity estimate for the linearized quantum Boltzmann-Bose operator with angular singularity, enabling analysis of stability and well-posedness of the quantum Boltzmann-Bose equation near equilibrium.

Contribution

It provides a constructive, explicit coercivity estimate that accounts for angular singularity and fugacity effects, advancing the mathematical understanding of the quantum Boltzmann-Bose equation.

Findings

Explicit coercivity estimate depending on fugacity

Global well-posedness near equilibrium established

Stability of Bose-Einstein equilibrium confirmed

Abstract

The quantum Boltzmann-Bose equation describes a large system of Bose-Einstein particles in the weak-coupling regime. If the particle interaction is governed by the inverse power law, the corresponding collision kernel has angular singularity. In this paper, we give a constructive proof of the coercivity estimate for the linearized quantum Boltzmann-Bose operator to capture the effects of the singularity and the fugacity. Precisely, the estimate explicitly reveals the dependence on the fugacity parameter before the Bose-Einstein condensation. With the coercivity estimate, the global in time well-posedness of the inhomogeneous quantum Boltzmann-Bose equation in the perturbative framework and stability of the Bose-Einstein equilibrium can be established.