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

• DOI: 10.1007/s00205-010-0290-1
• Publication Date: 2010-01-27T08:50:21Z
• Authors: Radjesvarane Alex...; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
• Keywords: pseudo-differential calculus; 35B65; Non-cutoff cross-sections; 35A05, 35B65; 01 natural sciences; 35A05; 510; Boltzmann equation; uncertainty principle; Mathematics - Analysis of PDEs; FOS: Mathematics; regularizing effect; [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]; 0101 mathematics; local existence; Analysis of PDEs (math.AP)

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far 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 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 Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution enjoys the $C^\infty$ regularity for positive time.