Topics in Kinetic Theory

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Kinetic theory provides a description of the dynamics of a large system of interacting particles by considering the particle density function instead of tracking each particle separately. Such averaging approach was first used to derive the Boltzmann equation that models the evolution of rarefied gases with predominantly binary interactions, and it has been successfully applied to derive kinetic equations modeling a wide range of applications, from gas and plasma dynamics to high atmosphere aerodynamics and collective behavior in social networks. This project will focus on two types of kinetic equations. The first type is a kinetic model which goes beyond binary interactions via incorporating a sum of higher order interaction terms, and as such serves as a basic model for a non-ideal gas. The second type of equations to be considered model dilute hot plasmas, which appear in nuclear fusion and tokamaks.The goal of this project is to advance the mathematical understanding of nonlinear partial differential equations arising in kinetic theory. The first part of the project focuses on questions of well-posedness, moment estimates, and description of long-range interactions of equations that generalize the Boltzmann equation. One such generalization is the binary-ternary Boltzmann equation that models the evolution of denser gases which allow both binary and ternary interactions. The second part considers the long-time behavior, including the rate of convergence to the equilibrium, moment estimates, and propagation of smoothness versus formation of blow up in finite time, of the relativistic Landau equation. This equation captures the effects of Einstein’s’ theory of special relativity due to high particle velocities in hot plasmas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.