Applied Mathematics Seminar

Title: Exponential tails for the non-cutoff Boltzmann equation

Speaker: Maja Taskovich (UPenn) 
https://www.math.upenn.edu/~taskovic/

Abstract: The Boltzmann equation models the evolution of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting.

We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments, which can be understood as a generalization of exponential moments. An interesting aspect of the result is that the singularity rate of the angular kernel affects the order of tails that can be propagated in time. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.