The Boltzmann equation : sharp Povzner inequalities applied to regularity theory and Kaniel & Shinbrot techniques applied to inelastic existence
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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.