Hypocoercive diffusion operators

International Congress of Mathematicians, ICM 2006

Volumn 3, Issue , 2006, Pages 473-498

  Villani, Cédric ^a  

^a UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

Author keywords

Diffusion equations; Fokker Planck and Boltzmann equations; H Theorem; Hypocoercivity; Hypoellipticity; Logarithmic Sobolev inequalities; Spectral gap

Indexed keywords

DIFFUSION EQUATIONS; FOKKER PLANCK; H-THEOREM; HYPOCOERCIVITY; HYPOELLIPTICITY; LOGARITHMIC SOBOLEV INEQUALITY; SPECTRAL GAP;

PARTIAL DIFFERENTIAL EQUATIONS;

BOLTZMANN EQUATION;

EID: 78651089475     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (32)

1. Cáceres, M. J., Carrillo, J. A., and Goudon, T., Equilibration rate for the linear inhomogeneous relaxation-timeBoltzmann equation for charged particles.Comm.Partial Differential Equations 28 (5-6) (2003), 969-989.
2. Carrillo, J. A., McCann, R. J., and Villani, C., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (3) (2003), 971-1018.
3. Desvillettes, L., and Villani, C., 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.
4. Desvillettes, L., and Villani, C., On a variant of Korn's inequality arising in statistical mechanics. ESAIM Control Optim. Calc. Var. 8 (2002), 603-619.
5. Desvillettes, L., and Villani, C., On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159 (2) (2005), 245-316.
6. Eckmann, J.-P., and Hairer, M., Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Comm. Math. Phys. 212 (1) (2000), 105-164.
7. Eckmann, J.-P., and Hairer, M., Spectral properties of hypoelliptic operators. Comm. Math. Phys. 235 (2) (2003), 233-253.
8. Eckmann, J.-P., Pillet, C.-A., and Rey-Bellet, L., Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Comm. Math. Phys. 201 (3) (1999), 657-697.
9. Fellner, K., Neumann, L., and Schmeiser, C., Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes. Monatsh. Math. 141 (4) (2004), 289-299.
10. 2 N. SIAM J. Sci. Comput., to appear.
11. Gallay, T., and Wayne, C. E., Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Comm. Math. Phys. 255 (1) (2005), 97-129.
12. Guo, Y., The Landau equation in a periodic box. Comm. Math. Phys. 231 (3) (2002), 391-434.
13. Guo, Y., and Strain, R., Almost exponential decay near Maxwellian. Comm. Partial Differential Equations 31 (3) (2006), 417-429.
14. Guo, Y., and Strain, R., Exponential decay for soft potentials near Maxwellian. Preprint, 2005.
15. Hairer, M., and Mattingly, J., Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math, to appear.
16. Helffer, B., and Nier, F., Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Math. 1862, Springer-Verlag, Berlin 2005.
17. Hérau, F., Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal., to appear; http://helios.univreims.fr/Labos/Mathematiques/Homepages/Herau/.
18. Hérau, F., Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. Preprint (revised version), 2005; http://helios.univreims.fr/Labos/Mathematiques/Homepages/Herau/.
19. Hérau, F., and Nier, F., Isotropic hypoellipticity and trend to equilibrium for the Fokker- Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171 (2) (2004), 151-218.
20. Hörmander, L., Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147-171.
21. Mouhot, C., and Neumann, L., Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Preprint, 2006.
22. Rey-Bellet, L., and Thomas, L. E., Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Comm. Math. Phys. 215 (1) (2000), 1-24.
23. Rey-Bellet, L., and Thomas, L. E., Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225 (2) (2002), 305-329.
24. Talay, D., 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.
25. Toscani, G., and Villani, C., 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.
26. Toscani, G., and Villani, C., On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys. 98 (5-6) (2000), 1279-1309.
27. Villani, C., A review of mathematical topics in collisional kinetic theory. In Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam 2002, 71-305.
28. Villani, C., Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234 (3) (2003), 455-490.
29. Villani, C., Entropy dissipation and convergence to equilibrium. Notes from a series of lectures at Institut Henri Poincaré, Paris 2001 (updated 2004); http://www.umpa.enslyon.fr/~cvillani/.
30. Villani, C., 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 .
31. Villani, C., Hypocoercive diffusion operators in Hörmander form. Preprint, 2006; http://www.umpa.ens-lyon.fr/~cvillani/.
32. Villani, C., Hypocoercive nonlinear diffusion operators. Preprint, 2006.