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Mathematics Colloquium – Dr. Maja Taskovic

Author: las-digital

Exponential tails for the Boltzmann equation

In kinetic theory, a large system of interacting particles is described by a particle probability distribution function.
One of the first equations derived in such a way was the Boltzmann equation (derived by Maxwell in 1866 and by
Boltzmann in 1872). 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 with a belief that such an assumption does not affect the
equation significantly. However, in 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 the 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, and Pavlovic.