Exponential convergence to equilibrium of solutions of the Kac equation and homogeneous Boltzmann equation for Maxwellian without angular cut-off

For the Kac equation and homogeneous Boltzmann equation ofMaxwellian without Grad's angular cut-off, we prove an exponential convergence towards the equilibrium as t -> infinity in a weak norm which is equivalent to the weak convergence of measures, extending results of Gabetta, Toscani and Wennberg (J. Stat. Phys. 81 (1995), 901-934) and Carlen, Gabetta and Toscani (Commun. Math. Phys. 199 (1999), 521-546) from the cut-off case to the non-cut-off case. We give quantitative estimates of the convergence rate, which are governed by the spectral gap of the linearized collision operator. We then prove a uniform bound in time on Sobolev norms of the solutions. The results are then combined with some interpolation inequalities, to obtain the rate of the exponential convergence in the strong L-1 norm, as well as various Sobolev norms.