For the homogeneous Boltzmann equation with (cutoo or non cutoo) hard potentials, we prove estimates of propagation of L p norms with a weight (1+jxj 2) q=2 (1 < p < +1, q 2 R large enough), as well as appearance of such weights. The proof is based on some new functional inequalities for the collision operator, proven by elementary means.

We apply the spectral-Lagrangian method of Gamba and Tharkabhushanam for solving homogeneous Boltzmann equation to compute low probability tails velocity distribution function, f, a particle species. This is based on truncation, Qtr(f,f), collision operator, Q(f,f), whose Fourier transform given by weighted convolution. The truncated operator models situation in which two colliding particles ig...