Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

International audienceThe Boltzmann equation without Grad’s angular cutoff assumption is believedto have a regularizing effect on the solutions because of the non-integrable angularsingularity of the cross-section. However, even though this has been justifiedsatisfactorily for the spatially homogeneous Boltzmann equation, it is still basicallyunsolved for the spatially inhomogeneous Boltzmann equation. In this paper,by sharpening the coercivity and upper bound estimates for the collision operator,establishing the hypo-ellipticity of the Boltzmann operator based on a generalizedversion of the uncertainty principle, and analyzing the commutators between thecollision operator and some weighted pseudo-differential operators, we prove theregularizing effect in all (time, space and velocity) variables on the solutions whensome mild regularity is imposed on these solutions. For completeness, we also showthat when the initial data has this mild regularity and a Maxwellian type decay inthe velocity variable, there exists a unique local solution with the same regularity,so that this solution acquires the C∞ regularity for any positive time