Abstract:
The Kac equation is a model for a one-dimensional rarefied gas in
which particle collisions conserve mass and energy but not the momentum. It
can be viewed as a simplified model of the Boltzmann equation for Maxwell
molecules. In 1993 Desvillettes proved that the homogeneous Kac equation
with an angular cutoff propagates exponential moments of order 2 and of
order 1. We apply recent techniques of Mittag-Leffler moments to show
propagation of exponential moments of all orders between 0 and 2 (not
necessarily integers). Using the same technique we are also able to treat
the non-cutoff Kac equation, in which case the order of the exponential
moment depends on the singularity rate of the angular kernel. We also
consider the Boltzmann equation for Maxwell molecules, both with and
without an angular cutoff. This is a joint work with Milana Pavic-Colic. |