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A remark on the generalized Toscani metric in probability measures with moments. (English) Zbl 1404.35315

Summary: Motivated by a pioneer work of H. Tanaka [Z. Wahrscheinlichkeitstheor. Verw. Geb. 46, 67–105 (1978; Zbl 0389.60079)] 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 Mawellian molecules, introducing the so-called Toscani metric on probability measures with moments less than 2; in [Commun. Pure Appl. Math. 63, No. 6, 747–778 (2010; Zbl 1205.35180)], M. Cannone and G. Karch studied infinite energy solutions to the above Cauchy problem, which include self-similar solutions given by A. V. Bobylev and C. Cercignani [J. Stat. Phys. 106, No. 5–6, 1039–1071 (2002; Zbl 1001.82091)]. The existence result of Cannone and Karch [loc. cit.] for the mild singular cross section of the Boltzmann collision term was extended to the strong singular case by Y. Morimoto [Kinet. Relat. Models 5, No. 3, 551–561 (2012; Zbl 1255.35177)], and the smoothing effect for measure valued (finite and/or infinite energy) solutions has been completely solved in [the second and the last authors, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32, No. 2, 429–442 (2015; Zbl 1321.35125); the second author et al., J. Math. Pures Appl. (9) 103, No. 3, 809–829 (2015; Zbl 1308.35155); Anal. Appl., Singap. 15, No. 3, 391–411 (2017; Zbl 1368.35202)] (see also [the second author et al., J. Stat. Phys. 165, No. 5, 866–906 (2016; Zbl 1360.35133)]for the non-Maxwellian molecules case). In [Morimoto and Yang, loc. cit.; Morimoto et al., loc. cit.], the Toscani metric was generalized in order to characterize perfectly the Fourier image of probability measures with moments less than 2. Furthermore, in [SIAM J. Math. Anal. 48, No. 4, 2399–2413 (2016; Zbl 1347.35189)], the 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 transform, simplifying an earlier work [“Absolute moments and Fourier-based probability metrics”, Preprint, arXiv:1510.08667] by the first author, where the forward difference operator and its iteration are used. This simple generalized Toscani metric was applied in [the authors, 2016, loc. cit.] to show the continuity of the solution in \(L^1_\alpha\) with respect to any positive time when the initial measure datum possesses finite moment of order \(\alpha >2\), 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 equation, after a short review about the research on measure valued solutions to the spatially homogeneous non-cutoff Boltzmann equation of Maxwellian molecules.

MSC:

35Q20 Boltzmann equations
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35H20 Subelliptic equations
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B40 Asymptotic behavior of solutions to PDEs
35R11 Fractional partial differential equations
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