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The Boltzmann equation is studied without the cutoff assumption. Under aperturbative setting, a unique global solution of the Cauchy problem of theequation is established in a critical Chemin-Lerner space. In order to analysethe collisional term of the equation, a Chemin-Lerner norm is combined with anon-isotropic norm with respect to a velocity variable, which yields an aprioriestimate for an energy estimate. Together with local existence following fromcommutator estimates and the Hahn-Banach extension theorem, the desiredsolution is obtained. Also, the non-negativity of the solution is rigorouslyshown.

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