Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates

We provide a new analysis of the Boltzmann equation with a constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is L-2x,v(2); we prove the global well-posedness and a version of scattering, assuming that the data f(0) is sufficiently smooth and localized, and the L-x,v(2) norm of f(0) is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision "gain" term in Boltzmann's equation, combined with a novel application of the Kaniel-Shinbrot iteration.