Exponential Convergence to Equilibrium for the Homogeneous Boltzmann Equation for Hard Potentials Without CutOff
Abstract
This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the socalled (non cutoff and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani from \cite{Vill1} where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a $L^1$ space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani and al in \cite{GMM}. We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.
 Publication:

Journal of Statistical Physics
 Pub Date:
 November 2014
 DOI:
 10.1007/s109550141066z
 arXiv:
 arXiv:1403.7619
 Bibcode:
 2014JSP...157..474T
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 22 pages