Abstract:  In the context of the Boltzmann equation, functional inequalities relating entropy dissipation and relative entropy to equilibrium are fundamental to obtaining explicit rates of relaxation to equilibrium.

In this talk, I present a method of transfer of inequalities, which establishes an (almost) equivalence, regarding entropy inequalities, between the classical and the fermionic Boltzmann cases. We thus obtain a large class of such inequalities in the fermionic case, and therefore, quantitative relaxation rates towards equilibrium for solutions to the (homogeneous cut-off hard potentials) Boltzmann-Fermi-Dirac equation.

Abstract:  In this seminar, we will briefly describe some variants of a kinetic model for the anomalous response of the immune system. These models encompass cell proliferation and destruction, along with other effects relevant to autoimmune diseases. We then consider the corresponding macroscopic models and investigate various mathematical properties of both kinetic and macroscopic systems, including equilibrium states, stability, relapse-remission patterns, and time-delay effects. Finally, we will present and interpret some numerical results to illustrate the model's capabilities in the biological context.

Abstract:  We study equilibration rates for nonlocal Fokker-Planck equations arising in swarm manufacturing. The PDEs of interest possess a time-dependent nonlocal diffusion coefficient and a nonlocal drift, modeling the interaction of a large system of agents. The emerging steady profile is characterized by a uniform spreading over a portion of the domain. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution with the properties of the quasi-stationary profile.

Abstract:  The traditional equations of fluid dynamics with the constitutive laws of Navier-Stokes and Fourier for stress tensor and heat flux are known to lose their validity when the Knudsen number - the ratio between the mean free path and a macroscopic length - becomes significantly large. Instead, the non-equilibrium regime requires modeling based on the statistical description of kinetic gas theory and Boltzmann equation. Using moment equations we extend the classical fluid dynamic equations for processes with moderate Knudsen numbers. We will present the regularized 13-moment-equations (R13) within the framework of moment approximations which offer a compromise between stability and robustness, simplicity and physical accuracy. We will discuss the mathematical structure and physical predictions of the R13 system, as well as numerical solutions in detail.

Abstract:  The aim of this lecture is to present some recent results on the physical intuition of the polyatomic Boltzmann operator in the continuous setting for the internal energy. The idea is to evaluate Boltzmann operator and extract models for transport coefficients. These models contain parameters of the collision kernel. We discuss possibility of matching those parameters with the experimental data, first for the collision kernel used in the rigorous analysis endowed with frozen collisions, and then for the extended model of the collision kernel. A possible extension to the mixture of Eulerian fluids will be discussed as well.