Kinetic and related models with applications in the natural sciences

The Cauchy problem and BEC stability for the quantum Boltzmann-Gross-Pitaevskii system

Irene Gamba

University of Texas at Austin

We rigorously discuss the wellposedness of the system that describes the dynamics of a Bose-Einstein Condensate consists of a quantum Boltzmann equation of the excitation distribution function and the Gross-Pitaevskii equation of the condensate wave function. The problems consistes the Cauchy problem for the quantum Boltzmann equation, that approximates the evolution of the distribution function of the excitation-thermal cloud, at the temperature regime which is very low compared to the Bose-Einstein Condensation critical temperature. Such an equation has a cubic kinetic transition probability kernel. We develop the existence and uniqueness result by means of abstract ODE's theory in Banach spaces by characterizing an invariant bounded, convex, closed subset of the positive cone associated with the Banach space of integrable densities with all polynomial moments (observables) globally bounded. In addition, we show the scaled summability of polynomial moments by studying the propagation and generation of exponential moments characterizing the high energy tails of the probability density solutions. This is work in collaboration with Ricardo Alonso and Minh Binh Tran. In addition this problem has inspired the recent work on constructions of solutions for the Wave turbulence kinetic models (work in collaboration with Leslie Smith and Minh Binh Tran, presented in this conference as well). discuss some similarities and differences that will be shown, in the M.B.Tran's lecture, yield a quite different long time behavior.