Convolution inequalities for the Boltzmann collision operator - Mathematics > Analysis of PDEsReport as inadecuate




Convolution inequalities for the Boltzmann collision operator - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: We study integrability properties of a general version of the Boltzmanncollision operator for hard and soft potentials in $n$-dimensions. Areformulation of the collisional integrals allows us to write the weak form ofthe collision operator as a weighted convolution, where the weight is given byan operator invariant under rotations. Using a symmetrization technique in$L^p$ we prove a Young-s inequality for hard potentials, which is sharp forMaxwell molecules in the $L^2$ case. Further, we find a newHardy-Littlewood-Sobolev type of inequality for Boltzmann collision integralswith soft potentials. The same method extends to radially symmetric,non-increasing potentials that lie in some $L^{s} {weak}$ or $L^{s}$. Themethod we use resembles a Brascamp, Lieb and Luttinger approach for multilinearweighted convolution inequalities and follows a weak formulation setting.Consequently, it is closely connected to the classical analysis of Young andHardy-Littlewood-Sobolev inequalities. In all cases, the inequality constantsare explicitly given by formulas depending on integrability conditions of theangular cross section in the spirit of Grad cut-off. As an additionalapplication of the technique we also obtain estimates with exponential weightsfor hard potentials in both conservative and dissipative interactions.



Author: Ricardo J. Alonso, Emanuel Carneiro, Irene M. Gamba

Source: https://arxiv.org/







Related documents