DECAY ESTIMATES FOR LARGE VELOCITIES IN THE BOLTZMANN EQUATION WITHOUT CUTOFF

We consider solutions f = f(t, x, v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x is an element of T-d, for hard and moderately soft potentials without the angular cutoff assumption, and under the a priori assumption that the main hydrodynamic fields, namely the local mass integral f dv and local energy integral f vertical bar v vertical bar(2) dv and local entropy integral f ln f dv, are controlled along time. We establish quantitative estimates of propagation in time of pointwise polynomial moments, i.e., sup(x,v) f(t, x, v)(1 + vertical bar v vertical bar)(q), q >= 0. In the case of hard potentials, we also prove appearance of these moments for all q >= 0. In the case of moderately soft potentials, we prove the appearance of low-order pointwise moments. All these conditional bounds are uniform as t goes to +infinity, conditionally to the bounds on the hydrodynamic fields being uniform.