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

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 linearised operator. This result improves the polynomial in time convergence obtained by Desvillettes and Villani [5]. Our approach is based on new decay estimates for the semigroup generated by the linearised Landau operator in weighted (polynomial or stretched exponential) L^p-spaces, using a method developed by Gualdani, Mischler and Mouhot [7].