Communications in Mathematical Physics

, Volume 298, Issue 2, pp 293-322

Convolution Inequalities for the Boltzmann Collision Operator

  • Ricardo J. AlonsoAffiliated withDept. of Computational & Applied Mathematics, Rice University
  • , Emanuel CarneiroAffiliated withSchool of Mathematics, Institute for Advanced Study
  • , Irene M. GambaAffiliated withDepartment of Mathematics, University of Texas at Austin Email author 

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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 L p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L 2 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 L s . 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.