Archive for Rational Mechanics and Analysis

, Volume 198, Issue 1, pp 39–123

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

  • Radjesvarane Alexandre
  • Yoshinori Morimoto
  • Seiji Ukai
  • Chao-Jiang Xu
  • Tong Yang
Article

DOI: 10.1007/s00205-010-0290-1

Cite this article as:
Alexandre, R., Morimoto, Y., Ukai, S. et al. Arch Rational Mech Anal (2010) 198: 39. doi:10.1007/s00205-010-0290-1

Abstract

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.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Radjesvarane Alexandre
    • 1
  • Yoshinori Morimoto
    • 2
  • Seiji Ukai
    • 3
  • Chao-Jiang Xu
    • 4
    • 5
  • Tong Yang
    • 6
  1. 1.IRENAV Research InstituteFrench Naval AcademyBrest-LanvéocFrance
  2. 2.Graduate School of Human and Environmental StudiesKyoto UniversityKyotoJapan
  3. 3.Hodogaya-ku, YokohamaJapan
  4. 4.Université de Rouen, UMR 6085-CNRS MathématiquesSaint-Etienne du RouvrayFrance
  5. 5.School of MathematicsWuhan UniversityWuhanChina
  6. 6.Department of MathematicsCity University of Hong KongHong KongPeople’s Republic of China