Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials

Kleber Carrapatoso (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper deals with the long time behaviour of solutions to the spatially homogeneous Landau equation with hard potentials . We prove an exponential in time convergence towards the equilibrium with the optimal rate given by the spectral gap of the associated linearized operator. This result improves the polynomial in time convergence obtained by Desvillettes and Villani \cite{DesVi2}. Our approach is based on new decay estimates for the semigroup generated by the linearized Landau operator in weighted (polynomial or stretched exponential) $L^p$-spaces, using a method develloped by Gualdani, Mischler and Mouhot \cite{GMM}.

Mots clés: hard potentials; hypodissipativity; exponential decay; Landau equation; spectral gap

Domaines: Equations aux dérivées partielles [math.AP]

Kleber Carrapatoso. Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials. Bulletin des Sciences Mathématiques,, 2015, 139, pp.777-805. ⟨10.1016/j.bulsci.2014.12.002⟩. ⟨hal-00851757v4⟩