Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.