Speaker: Maria Gualdani, George Washington University

Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936 Lev Landau derived from the Boltzmann equation a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau and Boltzmann equations is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. The Boltzmann and Landau equations present integro-differential operators that are highly nonlinear, singular and with degenerating coefficients. Despite the fact that many mathematicians and physicists have been working on these equations, many important questions are still unanswered due to their mathematical complexity. In this talk we concentrate on the mathematical results of the Landau equation. We will first review existing results and open problems and in the second part of the talk we will focus on recent developments of well-posedness and regularity theory.