Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin–Lerner space. In order to analyze the collisional term of the equation, a Chemin–Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn–Banach extension theorem, the desired solution is obtained.