A remark on the generalized Toscani metric in probability measures with moments (Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics)

Abstract

Motivated by a pioneer work of Hiroshi Tanaka[22](1978) by means of the probabilistic method, from the middle of 1990s, Toscani and his coauthors analytically studied the existence, the uniqueness and the asymptotic behavior of solutions to the Cauchy problem for the non cutoff spatially homogeneous Boltzmann equation of Maxwellian molecules, introducing the so-called Toscani metric defined in the space of the Fourier image of probability measures. By using the Toscani metric on probability measures with moments less than 2, Cannone-Karch[5] studied infinite energy solutions to the above Cauchy problem, which include self-similar solutions given by Bobylev-Cercignani[4]. The existence result of [5] for the mild singular cross section of the Boltzmann collision term was extended to the strong singular case by [15], and the smoothing effect for measure valued (finite and/or infinite energy) solutions has been completely solved in [19, 16, 17] (see also [18] for the non-Maxwellian molecules case). In [16, 17], the Toscani metric was generalized in order to characterize perfectly the Fourier image of probability measures with moments less than 2. Furthermore, in [9] authors have characterized the class of probability measures possessing finite moments of any positive order, in terms of the symmetric difference operators of their Fourier transforms, simplifying an earlier work[8] by the first author, where the forward difference operator and its iteration are used. This simple generalized Toscani metric was applied in [9] to show the continuity of the solution in L1 α with respect to any positive time when the initial measure datum posses finite moment of order α > 2, implicitly based on the equivalence between the generalized Toscani metric and the Monge-Kantorovich-Wasserstein metric. The purpose of this note is to give a supplementary proof of this equivalence, after a short review about the research on measure valued solutions to the spatially homogeneous non-cutoff Boltzmann equation of Maxwellian molecules.