Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off

Abstract

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off 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.

22 pages

References in corpus (3)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Spectral analysis of semigroups and growth-fragmentation equations
• Fractional Fokker-Planck equation

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• Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off
• Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties
• On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials
• Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials