UPPER BOUNDS FOR THE FUNCTION SOLUTION OF THE HOMOGENEOUS 2D BOLTZMANN EQUATION WITH HARD POTENTIAL

We deal with f(t)(dv), the solution of the homogeneous 2D Boltzmann equation without cutoff. The initial condition f(0)(dv) may be any probability distribution (except a Dirac mass). However, for sufficiently hard potentials, the semigroup has a regularization property (see Probab. Theory Related Fields 151 (2011) 659-704): f(t)(dv) = f(t)(v) dv for every t > 0. The aim of this paper is to give upper bounds for f(t)(v), the most significant one being of type f(t)(v) <= Ct(-eta)e(-vertical bar v vertical bar lambda) for some eta, lambda > 0.