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

Isabelle Tristani (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Boltzmann equation without cut-off; hard potentials; spectral gap; dissipativity; exponential rate of convergence; long-time asymptotic

Domaines: Equations aux dérivées partielles [math.AP]

Isabelle Tristani. Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off. 2014. ⟨hal-00967564v2⟩