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We provide a new analysis of the Boltzmann equation with constant collisionkernel in two space dimensions. The scaling-critical Lebesgue space is$L^2_{x,v}$; we prove global well-posedness and a version of scattering,assuming that the data $f_0$ is sufficiently smooth and localized, and the$L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a newscaling-critical bilinear spacetime estimate for the collision "gain" term inBoltzmann's equation, combined with a novel application of the Kaniel-Shinbrotiteration.

Small data global well-posedness for a Boltzmann equation via bilinear spacetime estimates