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In recent decades, kinetic theory - originally developed as a field of mathematical physics - has emerged as one of the most prominent fields of modern mathematics. In recent years, there has been an explosion of applications of kinetic theory to other areas of research, such as biology and social sciences. This book collects lecture notes and recent advances in the field of kinetic theory of lecturers and speakers of the School “Trails in Kinetic Theory: Foundational Aspects and Numerical Methods”, hosted at Hausdorff Institute for Mathematics (HIM) of Bonn, Germany, 2019, during the Junior Trimester Program “Kinetic Theory”. Focusing on fundamental questions in both theoretical and numerical aspects, it also presents a broad view of related problems in socioeconomic sciences, pedestrian dynamics and traffic flow management.



Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods

Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review Villani (J Stat Phys 124(2):781–822, 2006) by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods.
José Antonio Carrillo, Jingwei Hu, Zheng Ma, Thomas Rey

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

We consider kinetic and associated macroscopic equations on networks. A general approach to derive coupling conditions for the macroscopic equations from coupling conditions of the underlying kinetic problem is presented using an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. The procedure is explained for two simplified situations. The linear case is discussed for a linear kinetic BGK-type model leading in the macroscopic limit to a linear hyperbolic problem. The nonlinear situation is investigated for a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic conservation law on a network. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks.
Raul Borsche, Axel Klar

Coagulation Equations for Aerosol Dynamics

Binary coagulation is an important process in aerosol dynamics by which two particles merge to form a larger one. The distribution of particle sizes over time may be described by the so-called Smoluchowski’s coagulation equation. This integrodifferential equation exhibits complex non-local behaviour that strongly depends on the coagulation rate considered. We first discuss well-posedness results for the Smoluchowski’s equation for a large class of coagulation kernels as well as the existence and nonexistence of stationary solutions in the presence of a source of small particles. The existence result uses Schauder fixed point theorem, and the nonexistence result relies on a flux formulation of the problem and on power law estimates for the decay of stationary solutions with a constant flux. We then consider a more general setting. We consider that particles may be constituted by different chemicals, which leads to multi-component equations describing the distribution of particle compositions. We obtain explicit solutions in the simplest case where the coagulation kernel is constant by using a generating function. Using an approximation of the solution we observe that the mass localizes along a straight line in the size space for large times and large sizes.
Marina A. Ferreira

Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint

These lecture notes address mathematical issues related to the modeling of impact laws for systems of rigid spheres and their macroscopic counterpart. We analyze the so-called Moreau’s approach to define multibody impact laws at the mircroscopic level, and we analyze the formal macroscopic extensions of these laws, where the non-overlapping constraint is replaced by a barrier-type constraint on the local density. We detail the formal analogies between the two settings, and also their deep discrepancies, detailing how the macroscopic impact laws, natural ingredient in the so-called Pressureless Euler Equations with a Maximal Density Constraint, are in some way irrelevant to describe the global motion of a collection of inertial hard spheres. We propose some preliminary steps in the direction of designing macroscopic impact models more respectful of the underlying microscopic structure, in particular we establish micro-macro convergence results under strong assumptions on the microscopic structure.
Félicien Bourdin, Bertrand Maury

An Introduction to Uncertainty Quantification for Kinetic Equations and Related Problems

We overview some recent results in the field of uncertainty quantification for kinetic equations and related problems with random inputs. Uncertainties may be due to various reasons, such as lack of knowledge on the microscopic interaction details or incomplete information at the boundaries or on the initial data. These uncertainties contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. After a brief introduction on the main numerical techniques for uncertainty quantification in partial differential equations, we focus our survey on some of the recent progress on multi-fidelity methods and stochastic Galerkin methods for kinetic equations.
Lorenzo Pareschi

A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory

The content of these notes is based on a series of lectures given by the first author at HIM, Bonn, in May 2019. They provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases and weakly interacting plasmas respectively. These equations can be derived formally, under suitable scaling limits, taking classical particle systems as a starting point. A rigorous proof of this limiting procedure is difficult and still largely open. We discuss some mathematical problems arising in this context.
Mario Pulvirenti, Sergio Simonella

Statistical Description of Human Addiction Phenomena

We study the evolution in time of the statistical distribution of some addiction phenomena in a system of individuals. The kinetic approach leads to build up a novel class of Fokker–Planck equations describing relaxation of the probability density solution towards a generalized Gamma density. A qualitative analysis reveals that the relaxation process is very stable, and does not depend on the parameters that measure the main microscopic features of the addiction phenomenon.
Giuseppe Toscani

Boltzmann-Type Description with Cutoff of Follow-the-Leader Traffic Models

In this paper we consider a Boltzmann-type kinetic description of Follow-the-Leader traffic dynamics and we study the resulting asymptotic distributions, namely the counterpart of the Maxwellian distribution of the classical kinetic theory. In the Boltzmann-type equation we include a non-constant collision kernel, in the form of a cutoff, in order to exclude from the statistical model possibly unphysical interactions. In spite of the increased analytical difficulty caused by this further non-linearity, we show that a careful application of the quasi-invariant limit (an asymptotic procedure reminiscent of the grazing collision limit) successfully leads to a Fokker–Planck approximation of the original Boltzmann-type equation, whence stationary distributions can be explicitly computed. Our analytical results justify, from a genuinely model-based point of view, some empirical results found in the literature by interpolation of experimental data.
Andrea Tosin, Mattia Zanella
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