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Tanaka theorem for inelastic Maxwell models. (English) Zbl 1136.82033

Summary: We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [H. Tanaka, Z. Wahrsch. Verw. Geb. 46, 67–105 (1978; Zbl 0389.60079); C. Villani, Topics in optimal transportation, Amer. Math. Soc., Providence, RI (2003; Zbl 1106.90001)]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
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