Condensation and Cascades in Kinetic Models for Mass Exchange, Bose Gases, and Wave Turbulence

Abstract

This talk presents three spatially homogeneous kinetic models for mass exchange, Bose gases, and wave turbulence, with a focus on concentration and transfer across mass or energy scales.

The talk will begin with the Boltzmann equation with mass exchange introduced by P. Degond and J.-G. Liu. It will discuss existence and uniqueness results for weak solutions and explain the estimates that exclude concentration at zero mass. For a class of regularly varying mass-exchange rates, it will also present a mechanism that prevents finite-time gelation, namely the escape of mass to infinite particle size.

The talk will then turn to the Boltzmann equation for bosons. For a class of collision kernels, we prove finite-time Bose–Einstein condensation below the critical temperature without assuming initial concentration near zero energy. We also establish convergence to equilibrium, including convergence of the condensate mass.

Finally, the talk will present our results on finite-time condensation phase transition for quasilinear four-wave kinetic equations to explain how the interaction coefficient governs condensate formation at zero frequency. The lecture will include the main analytical ideas behind these results. This is joint work with Professor Jian-Guo Liu, Duke University.