Hypocoercive Diffusion Operators

Villani, Cédric

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 257-275 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

In many problems coming from mathematical physics, the association of a degenerate diffusion operator with a conservative operator may lead to dissipation in all variables and convergence to equilibrium. One can draw an analogy with the well-studied phenomenon of hypoellipticity in regularity theory, and actually both phenomena have been studied together. Now a distinctive theory of ``hypocoercivity'' is starting to emerge, with already some striking results, and several challenging open problems. This text (an abbreviated version of the one which I prepared for the International Congress of Mathematicians) will review some of them.

Résumé

In molti problemi provenienti dalla fisica matematica, l'associazione di un operatore di diffusione degenere con un operatore conservativo può portare a dissipazione in tutte le variabili e a convergenza verso l'equilibrio. Si può tracciare un'analogia con il fenomeno ben studiato di ipoellitticità nella teoria della regolarità, ed effettivamente entrambi i fenomeni sono stati studiati insieme. Ora una teoria distinta di ``ipocoercività'' sta iniziando ad emergere con alcuni risultati già sorprendenti e numerosi problemi aperti pieni di sfida. Questo testo (una versione abbreviata di quello che ho preparato per il Congresso Internazionale dei Matematici) ne analizza alcuni.

Détail

Publié le : 2007-06-01

MR 2339441 | Zbl 1178.35306

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