Weak Solutions of the Linear Boltzmann Equation with Very Soft Interactions

The Cauchy problem for the Boltzmann equation is treated for a class of interactions including inverse-power repulsive potentials Uρ = const · ρ−k+1, 95 < k ≤ 2 (in particular, the Coulomb potential) with some angular cutoff. Criteria are found for the existence of weak solutions to the linear spatially inhomogeneous Boltzmann equation with external forces. A weak solution is constructed as the weak limit of a sequence of mild solutions to the corresponding equations with suitably truncated scattering factors. Furthermore, strong convergence of weak solutions towards the equilibrium is established under detailed balance, using a generalized H-theorem, together with translation continuity.