Browsing by Subject "Kinetic theory of gases"

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The Boltzmann equation : sharp Povzner inequalities applied to regularity theory and Kaniel & Shinbrot techniques applied to inelastic existence
(2008-08) Alonso, Ricardo Jose, 1972-; Martínez Gamba, Irene, 1957-
This work consists of three chapters. In the first chapter, a brief overview is made on the history of the modern kinetic theory of elastic and dilute gases since the early stages of Maxwell and Boltzmann. In addition, I short exposition on the complexities of the theory of granular media is presented. This chapter has the objectives of contextualize the problems that will be studied in the remainder of the document and, somehow, to exhibit the mathematical complications that may arise in the inelastic gases (not present in the elastic theory of gases). The rest of the work presents two self-contained chapters on different topics in the study of the Boltzmann equation. Chapter 2 focuses in studying and extending the propagation of regularity properties of solutions for the elastic and homogeneous Boltzmann equation following the techniques introduced by A. Bobylev in 1997 and Bobylev, Gamba and Panferov in 2002. Meanwhile, chapter 3 studies the existence and uniqueness of the inelastic and inhomogeneous Cauchy problem of the Boltzmann equation for small initial data. A new set of global in time estimates, proved for the gain part of the inelastic collision operator, are used to implement the scheme introduced by Kaniel and Shinbrot in the late 70’s. This scheme, known as Kaniel and Shinbrot iteration, produces a rather simple and beautiful proof of existence and uniqueness of global solutions for the Boltzmann equation with small initial data.
Closures of the Vlasov-Poisson system
(2003) Jones, Christopher Scott; Morrison, Philip J.
Coerciveness, well-posedness, and Banach norms propagation of solutions to the system of Boltzmann equations for a monatomic gas mixture
(2021-12-02) De la Canal, Erica B.; Martínez Gamba, Irene, -1957; Pavić-Čolić, Milana; Caffarelli, Luis; Magin, Thierry; Pavlovic, Natasa; Vasseur, Alexis
In this thesis dissertation we focus on the Cauchy problem for the unique vector-valued solution to a system of full non-linear space homogeneous Boltzmann equations, describing multi-component monatomic gas mixtures with binary interactions in N-dimensions. The existence and uniqueness of such solution of the initial value problem, associated with a mixture in the case of hard potentials with integrable angular scattering kernels, is addressed by means of an abstract ODE theory in Banach spaces localized in the molecular velocity state. The initial data needs to be a vector of non-negative measures with finite total number density, momentum, and strictly positive energy, as well as to have finite L [superscript 1 over subscript k*] (ℝ [superscript N]) norms, referred to as the scalar polynomial moment of order k [subscript ∗], which is a sum of k [subscript ∗]-polynomial weighted norms for each species, based on the corresponding mass fraction parameter for each species as well as the intermolecular potential rates. The existence and uniqueness results rely on a lower bound for the collision frequency jointly with an energy identity for gas mixtures, which allows developing a new Angular Averaging Lemma for the gain part of the vector of collisional integrals with constants decaying with the corresponding dimensionless scalar polynomial moment order. The Angular Averaging Lemma combined with the hard potential hypothesis enables the generation and propagation of polynomial moments property for a vector-valued solution for any integrable angular transition, as well as its moments summability culminating in exponential moment estimates up to Gaussian tails. In addition, we extend propagation of such solution in the general vector-valued weighted L [superscript p](ℝ [superscript N]) Banach spaces. By means of a Carleman representation for the collision operator associated with binary interactions of any two gas mixture components we study L [superscript p](ℝ [superscript N]) integrability properties of the vector-valued multi-species collision operator gain term as a bilinear form. More specifically, we show gain of integrability estimates for the positive contribution of the multi-species collision operator providing explicit constant rates by means of controlling gain operator’s L [superscript p](ℝ [superscript N]) weighted norms by suitable sublinear L [superscript p](ℝ [superscript N]) weighted norm of the input function. The gain of integrability estimate together with the Lower bound introduced earlier, yields the propagation of L [superscript p](ℝ [superscript N]) norms with polynomial and/or exponential weights for p ∈ (1, ∞].