Global well-posedness of the quantum Boltzmann equation for bosons interacting via inverse power law potentials

Abstract

We consider the spatially inhomogeneous quantum Boltzmann equation for bosons with a singular collision kernel, the weak-coupling limit of a large system of Bose-Einstein particles interacting through inverse power law. Global well-posedness of the corresponding Cauchy problem is proved in a periodic box near equilibrium for initial data satisfying high temperature condition. Contents 1. Introduction 1 2. Linear operator estimate 12 3. Bilinear operator estimate 20 4. Trilinear operator estimate 31 5. Commutator estimate and weighted energy estimate 37 6. Local well-posedness 47 7. A priori estimate and global well-posedness 53 8. Appendix 60 References 69