Recent progress on global solutions to the homogeneous Landau equation.

The Landau equation is a fundamental kinetic model that describes the evolution of collisional plasmas. The equation includes a quadratic, non-local term that models the effects of binary collisions mediated by a Coulomb force. This collision term introduces substantial mathematical challenges, leaving many fundamental questions--such as the existence of global-in-time smooth solutions--largely open.

In this talk, I will explore recent progress made in understanding a simplified model, the homogeneous Landau equation, which retains the complex collision term. In a recent breakthrough work, Luis Silvestre and Nestor Guillen showed the existence of a new monotone functional---the Fisher information---which is used to construct global-in-time solutions for smooth rapidly decaying initial data. I will discuss joint work with Maria Gualdani and Amelie Loher, where we extend these results to general initial data and obtain new results on global-in-time existence and various forms of uniqueness. I will conclude with a discussion of how these results inform future research of the full model.