Abstract

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 a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau equation is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. Moreover, the coefficients are singular and degenerate for large velocities. Many important questions, such as whether or not solutions become unbounded after a finite time, are still unanswered due to their mathematical complexity. In this talk we concentrate on the mathematical results of the homogeneous Landau equation. We will first review existing results and open problems on global regularity versus blow-up in finite time. In the second part of the talk we will focus on recent developments of regularity theory for an isotropic version of the Landau equation.

This is a joint work with Nestor Guillen.