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Invited talk

Tuesday, July 20, 14:45 ~ 15:45 UTC-3

The Mathematics of Interacting Particle Systems by Boltzmann Type flows

Irene Matínez Gamba

The University of Texas at Austin, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The dynamics of Multilinear Interactive particle systems are statistically described by kinetic collisional modeling. Introduced in the last quarter of the nineteenth century by L. Boltzmann and J.C. Maxwell, independently, gave birth to the area of Mathematical Statistical Mechanics and Thermodynamics. These evolution models concern a class of non-local, and non-linear integro-differential operator equations of Boltzmann type, whose solutions are probability densities. These are dissipative systems. Their rigorous mathematical treatment and approximations are still emerging in comparison to classical non-linear PDE theory. All these models share a unified functional analysis framework identifying the role of scattering and partition functions as integrable forms over compact manifolds, endowed by a priori estimates generating coercivity as well as uniform bounds. A functional ODE in Banach spaces framework yields well-posedness in the Banach space in probability densities with moments, for well prepared weights, that induces dissipative effects and propagate entropy monotonicity. From there, it follows the propagation of $L^p_{2k}(\mathbb{R}^d)$-norms, $1\leq p \leq \infty$ and $k>k_*\ge 1$, where $k_*$ depends on the model's scattering mechanism, exponential moments, as well as higher Sobolev regularity theory. Their bounds can be traced back to the coercive constants and a priori moments upper bounds estimates. Applications range from the modeling of classical elastic billiard of single species of a monoatomic gas, to polyatomic gases interchanges, to system of particle species such as in mixtures with disparate masses, to low temperature regimes for quantum interactions, collisional plasmas or electron transport in nano structures, and to self-organized or social interacting dynamics.

Joint work with The most recent part of this work has been developed with Ricardo J. Alonso (Texas A&M - Qatar), Erica de la Canal (The University of Texas at Austin, USA), Milana Pavić-Čolić (Univerity of Novi Sad, Serbia; and RWTH Aachen University, Germany) and Maja Tascović (Emory University, USA).

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