Finite Time Blow-up of Solutions of Kinetic Equations and Formation of Bose-Einstein Condensate

Abstract

Some kinetic equations are known to blow up in finite time, although the detailed analysis of this phenomenon is generally quite difficult. It has long been suspected that the kinetic equation for a Bose gas, first derived by Nordheim (and called later on Boltzmann-Nordheim) has the property of blowing-up in finite time if the initial solutions is dense enough, compared to its energy, a phenomenon that could be viewed as a dynamical equivalent of the equilibrium Bose-Einstein condensation for a Bose gas at very low temperature. However it remains to show in a coherent way that, after this blow up some sort of condensate begins to grow. This article shows how that this actually happens thanks to the identification of the self similar equation for blow up as a non linear eigenvalue equation where one exponent acts as the non linear eigenvalue. This permits to relate the behavior of the solution of the kinetic equation before and after the blow up in a coherent way. Some consequences of this result for other kinetic equations are briefly presented.