Archive for Rational Mechanics and Analysis

, Volume 198, Issue 1, pp 39-123

First online:

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

  • Radjesvarane AlexandreAffiliated withIRENAV Research Institute, French Naval Academy
  • , Yoshinori MorimotoAffiliated withGraduate School of Human and Environmental Studies, Kyoto University
  • , Seiji UkaiAffiliated with Email author 
  • , Chao-Jiang XuAffiliated withUniversité de Rouen, UMR 6085-CNRS MathématiquesSchool of Mathematics, Wuhan University
  • , Tong YangAffiliated withDepartment of Mathematics, City University of Hong Kong

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The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved 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 generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C regularity for any positive time.