Natasa PAVLOVIC - On the Boltzmann equation in the non-cutoff case
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2016-07-24

The Boltzmann equation models the evolution of a rarefied gas, in which  particles interact through  binary  collisions,  by  describing  the  evolution    of  the  probability  density  of particles. The equation balances transport  operator with a collision operator, where the latter  is  a  bilinear    integral  with  an  angular  kernel  that  is  non-integrable  in  many  models. 
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,  it  has    recently  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  homogeneous 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.  We  show  how  the  singularity  rate  of    the 
angular kernel affects the order of the tails that can be propagated  in time.  
The talk is based on joint works with Alonso, Gamba, Taskovic and with Gamba, Taskovic.