MOMENT INEQUALITIES FOR THE BOLTZMANN-EQUATION AND APPLICATIONS TO SPATIALLY HOMOGENEOUS PROBLEMS

Some inequalities for the Boltzmann collision integral are proved. The se inequalities can be considered as a generalization of the well-know n Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equati on for a wide class of intermolecular forces. We obtain simple necessa ry and sufficient conditions (on the potential) for the uniform bounde dness of all moments. For potentials with compact support the followin g statement is proved: if all moments of the initial distribution func tion are bounded by the corresponding moments of the Maxwellian A exp( -Bv(2)), then all moments of the solution are bounded by the correspon ding moments of the other Maxwellian A(1) exp[-B-1(t) v(2)] for any t> 0; moreover B(t)=const for hard spheres. An estimate for a collision f requency is also obtained.