## Nonlinear Partial Differential Equations

**Date:** September 7, 2018

**Time:** 1:50PM - 2:50PM

**Location:** BLOC 628

**Speaker:** Irene Gamba, University of Texas

**Title:** *Nonlinear PDE's Seminar*

**Abstract:**

.Title: The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation System at very low temperature

Abstract: We discuss a quantum Boltzmann-Condensation system that describes the evolution of the interaction between a well formed Bose-Einstein Condensate (BEC) and the quasi-particles cloud. The kinetic model, derived as weak turbulence kinetic model from a quantum Hamiltonian, is valid for a dilute regime at which the temperature of a bosonic gas is very low compared to the Bose-Einstein condensation critical temperature. In particular, the system couples the density of the condensate from a Gross-Pitaevskii type equation to the kinetic equation through the dispersion relation in the kinetic model and the corresponding transition probability rate from pre to post collision momentum states.

We show the well-posedness of the Cauchy problem for the system, find qualitative properties of the solution such as instantaneous creation of exponential tails, and prove the uniform condensate stability related to the initial mass ratio between condensed particles and quasi-particles. This stability result leads to global in time existence of the initial value problem for the quantum Boltzmann-Condensation system.