Geodesic
Hypocoercive Diffusion Operators
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 2, pp. 257-275.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
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. 
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Villani, Cédric. Hypocoercive Diffusion Operators. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 2, pp. 257-275. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_2_a0/

[1] M. J. Cáceres - J. A. Carrillo - T. Goudon, Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. Partial Differential Equations 28 (5-6) (2003), 969-989. | DOI | MR

[2] J. A. Carrillo - R. J. Mccann - C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19 (3) (2003), 971-1018. | fulltext EuDML | DOI | MR | Zbl

[3] L. Desvillettes - C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (1) (2001), 1-42. | DOI | MR | Zbl

[4] L. Desvillettes - C. Villani, On a variant of Korn's inequality arising in statistical mechanics, ESAIM Control Optim. Calc. Var. 8 (2002), 603-619. | fulltext EuDML | DOI | MR | Zbl

[5] L. Desvillettes - C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2) (2005), 245-316. | DOI | MR | Zbl

[6] J.-P. Eckmann - M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. Math. Phys. 212 (1) (2000), 105-164. | DOI | MR | Zbl

[7] J.-P. Eckmann - M. Hairer, Spectral properties of hypoelliptic operators, Comm. Math. Phys. 235 (2) (2003), 233-253. | DOI | MR | Zbl

[8] J.-P. Eckmann - C.-A. Pillet - L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different tempera- tures, Comm. Math. Phys. 201 (3) (1999), 657-697. | DOI | MR | Zbl

[9] K. Fellner - L. Neumann - C. Schmeiser, Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes, Monatsh. Math. 141 (4) (2004), 289-299. | DOI | MR | Zbl

[10] F. Filbet - C. Mouhot - L. Pareschi, Solving the Boltzmann equation in $N \log_2 N$, SIAM J. Sci. Comput., to appear. | DOI | MR

[11] T. Gallay - C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys. 255 (1) (2005), 97-129. | DOI | MR | Zbl

[12] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (3) (2002), 391-434. | DOI | MR | Zbl

[13] Y. Guo - R. Strain, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, to appear. | DOI | MR | Zbl

[14] Y. Guo - R. Strain, Exponential decay for soft potentials near Maxwellian, is to appear in Arch. Rational Mech. Anal. | DOI | MR | Zbl

[15] M. Hairer - J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math, to appear. | DOI | MR | Zbl

[16] B. Helffer - F. Nier, Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Math. 1862, Springer-Verlag, Berlin 2005. | DOI | MR | Zbl

[17] F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., to appear; http://helios.univ-reims.fr/Labos/Mathematiques/Homepages/Herau/. | MR

[18] F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, Preprint (revised version), 2005; http://helios.univ-reims.fr/Labos/Mathematiques/Homepages/Herau/. | DOI | MR

[19] F. Hérau - F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2) (2004) | DOI | MR

[20] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. | DOI | MR

[21] C. Mouhot - L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, appeared in Nonlinearity, 19 (4) (2006), 969-998. | DOI | MR | Zbl

[22] L. Rey-Bellet - L. E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Comm. Math. Phys. 215 (1) (2000), 1-24. | DOI | MR | Zbl

[23] L. Rey-Bellet - L. E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys. 225 (2) (2002), 305-329. | DOI | MR | Zbl

[24] D. Talay, Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields 8 (2) (2002), 163-198. | MR | Zbl

[25] G. Toscani - C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys. 203 (3) (1999), 667-706. | DOI | MR | Zbl

[26] G. Toscani - C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds, J. Statist. Phys. 98 (5-6) (2000), 1279-1309. | DOI | MR | Zbl

[27] C. Villani, A review of mathematical topics in collisional kinetic theory. In Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam 2002, 71-305. | DOI | MR | Zbl

[28] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys. 234 (3) (2003), 455-490. | DOI | MR | Zbl

[29] C. Villani, Entropy dissipation and convergence to equilibrium. Notes from a series of lectures at Institut Henri Poincaré, http://www.umpa.ens-lyon.fr/~cvillani/, will appear in Lect. Notes in Math., Springer.

[30] C. Villani, Convergence to equilibrium: Entropy production and hypocoercivity. In Rarefied Gas Dynamics (ed. by M. Capitelli), AIP Conference Proceedings 762, American Institute of Physics, 2005, 8-25. | DOI | MR

[31] C. Villani, Hypocoercive diffusion operators, Preprint, 2006; http://www.umpa.ens-lyon.fr/~cvillani/. | MR | Zbl