Abstract

We will discuss in the fundamental functional features of scalar and systems of Boltzmann type flow. Binary interaction laws enable an existence and uniqueness theory for scalar or arbitrary systems modeling polyatomic gases or mixtures of gas particles with different masses, even as the BEC stability for quantum Boltzmann-condensation system models fell into a rather unified way after good moments and a priori estimates are obtained. From an analytical viewpoint, the crucial study lies in the identification of a Banach space associated to the interactions and their collision frequency characterized by the transition probability rates of hopping interacting particle pairs. This approach expands from the problem and initial data to yield upper and lower bounds and coerciveness that plays a crucial role in stability and longterm behavior of the system  arising from  solutions to ODEs in Banach spaces. In addition, the obtained estimates provide the ground to obtain a $W^{p,s}$- polynomial and exponentially weighted Sobolev regularity to solutions of these types of Boltzmann flows. These results do not depend on entropy estimates, yet if the initial entropy is bounded, then it will remain bounded depending on the initial invariant moments, for all times. Moreover, since these estimates yield sufficient conditions to control the high energy tail decay that sets a framework for the calculations to time decay rate to equilibrium for Boltzmann equation types, homogeneous in space, or with periodic boundary conditions, or with  boundary conditions corresponding to zero flux across boundaries. This is work in different collaborations with Ricardo Alonso, Erica de la Canal and Milana Pavic-Colic.