Irene M. Gamba

University of Texas at Austin

Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more

We will discuss the fundamental functional analysis features of Boltzmann type equations, that is, binary interaction laws that enables an existence and uniqueness theory for scalar or arbitrary systems modeling polyatomic gases or mixtures of gas particles with different masses respectively as much as BEC stability for quantum Boltzmann-condensation system models and more.

From an analytical viewpoint, it is important to identify a suitable Banach space associated to the interactions as much as to the collision frequency characterized by the transition probability rates of hopping interacting particle pairs. This approach expands from an initial Then, in within the functional space framework, we develop lower bounds that generate corciveness, and upper bounds for the Boltzmann bilinear form that enable these non-linear systems to admit a coercive constant to play the role of the Poincare constant in a classical Sobolev embedding argument. This coercive constant and upper bound not only is a venue of existence and uniqueness for Boltzmann flows by the construction of solutions to ODEs in a 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 type of Boltzmann flows.

These results do not depend on entropy estimates, yet if the initial entropy is bounded, then it will remain bounded by the initial one for all times. Moreover, 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 collaboration with in different in collaborations with Ricardo Alonso, Erica de la Canal, Milana Pavic-Colic, and Maja Taskovic.