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We study a quantum Boltzmann-Condensation system that describes the evolutionof the interaction between a well formed Bose-Einstein condensate and thequasi-particles cloud. The kinetic model is valid for a dilute regime at whichthe temperature of the gas is very low compared to the Bose-Einsteincondensation critical temperature. In particular, our system couples thedensity of the condensate from a Gross-Pitaevskii type equation to the kineticequation through the dispersion relation in the kinetic model and thecorresponding transition probability rate from pre to post collision momentumstates. We rigorously show the following three properties (1) thewell-posedness of the Cauchy problem for the system in the case of a radiallysymmetric initial configuration, (2) find qualitative properties of thesolution such as instantaneous creation of exponential tails and, (3) prove theuniform condensate stability related to the initial mass ratio betweencondensed particles and quasi-particles. The stability result from (3) leads toglobal in time existence of the initial value problem for the quantumBoltzmann-Condensation system.

The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation system for bosons at very low temperature