Penn Arts & Sciences Logo

Analysis Seminar

Thursday, February 6, 2025 - 3:30pm

Nestor Guillen

Texas State University

Location

University of Pennsylvania

DRL 4C4

Regularity for the Landau and Boltzmann equations via the Fisher information

In a recent work with Luis Silvestre we prove solutions to the (homogeneous) Landau equation do not blow up in finite time, settling a well known problem in kinetic theory. Blows up are ruled out by the finding that the Fisher information decreases over time along solutions of the equation. This finding is made possible by three ingredients: 1) a new “lifting procedure” involving a linear degenerate parabolic PDE in double the number of variables, 2) the rich symmetries of the lifted equation 3) a functional inequality closely related to the log-Sobolev inequality on the sphere. In a follow up work, Silvestre, Imbert, and Villani discover new integro-differential inequalities on the sphere allowing them to expand the monotonicity (and non blow up) results to the homogeneous Boltzmann equation with very soft potentials.  In this talk I will describe all three ingredients, motivating some of them first in the simpler case of the heat equation.

Other Events on This Day