Entropy dissipation and long-range interactions

We study Boltzmann's collision operator for long-range interactions, i.e., without Grad's angular cut-off assumption. We establish a functional inequa lity showing that the entropy dissipation controls smoothness of the distri bution function, in a precise sense. Our estimate is optimal, and gives a u nified treatment of both the linear and the nonlinear cases. We also give s imple and self-contained proofs of several useful results that were scatter ed in previous works. As an application, we obtain several helpful estimate s for the Cauchy problem, and for the Landau approximation in plasma physic s.