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The Landau Equation, Part 1 and Part 2 (copy)

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FKT - Frontiers in kinetic theory: connecting microscopic to macroscopic scales - KineCon 2022

Statistical physics originated after the development of molecular theory for matter. In molecular theory, gas, solid, or liquid matter are considered a collection of many identical interacting particles. Classical mathematical methods, such as differential equations that track the motion of each particle, became powerless for such large systems. Scientists ventured into statistical and probabilistic methods to describe large ensembles of objects. After the pioneering work of Bernoulli, Maxwell (1860) and Boltzmann (1868) used statistical physics to model the dynamics of gases. Their first investigations laid the foundation of kinetic theory and resulted in formulating the general equation of continuity, a partial differential equation known today as the Boltzmann equation. The Boltzmann equation is probabilistic: its solution represents the probability of finding a particle at time t at position x with velocity v. Later, in 1936, Lev Landau derived from the Boltzmann equation a new kinetic model to describe the motion of particles in hot plasma. From the 1960s, kinetic equations have been used to model dilute quantum particles that follow the Fermi-Dirac or Bose-Einstein statistics. This tutorial will encompass the existing mathematical theory of the Landau equation in both homogeneous and inhomogeneous settings. We will focus on one of the most challenging open questions, namely the global-in-time-regularity versus blow-up in finite time. We will also briefly introduce the Landau-Fermi-Dirac equation and present open problems. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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