Communications in Mathematical Physics

, Volume 298, Issue 2, pp 293–322

Convolution Inequalities for the Boltzmann Collision Operator

  • Ricardo J. Alonso
  • Emanuel Carneiro
  • Irene M. Gamba

DOI: 10.1007/s00220-010-1065-0

Cite this article as:
Alonso, R.J., Carneiro, E. & Gamba, I.M. Commun. Math. Phys. (2010) 298: 293. doi:10.1007/s00220-010-1065-0


We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in Lp we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L2 case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some \({L^{s}_{weak}}\) or Ls. The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Ricardo J. Alonso
    • 1
  • Emanuel Carneiro
    • 2
  • Irene M. Gamba
    • 3
  1. 1.Dept. of Computational & Applied MathematicsRice UniversityHoustonUSA
  2. 2.School of Mathematics, Institute for Advanced StudyPrincetonUSA
  3. 3.Department of MathematicsUniversity of Texas at AustinAustinUSA