Abstract
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.
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Communicated by H.-T. Yau
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Alonso, R.J., Carneiro, E. & Gamba, I.M. Convolution Inequalities for the Boltzmann Collision Operator. Commun. Math. Phys. 298, 293–322 (2010). https://doi.org/10.1007/s00220-010-1065-0
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DOI: https://doi.org/10.1007/s00220-010-1065-0