Powered by Indico
v3.2
Lecturer
Alessia Nota alessia.nota@gssi.it
Timetable and workload
Lectures: 24 hours
Course content
The derivation of effective macroscopic equations from microscopic descriptions based on the fundamental laws of mechanics—through suitable scaling limits—constitutes a central problem in non-equilibrium statistical mechanics, whose mathematical interest dates back to Hilbert's 6th problem.
In this course, we will address this question within the framework of kinetic theory of gases, focusing on two of the most notable kinetic equations: the Vlasov and Boltzmann equations. The Vlasov equation is a time-reversible transport equation that governs the statistical properties of plasmas and is widely used in cosmology and astrophysics to study the formation and evolution of galaxies. Conversely, the Boltzmann equation describes, at mesoscopic level, the irreversible dynamics of rarefied gases and has an extremely wide range of applications.
The goal of this introductory course is to present some key mathematical aspects of these prototype kinetic equations, highlighting established results related to their derivation from the dynamics of N-particle systems through suitable scaling limits. We will also provide an overview of important PDE results concerning their well-posedness theory and the asymptotic behavior of the solutions.
Recent progress, open questions, and possible research directions will be discussed towards the end of the course.