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# memo_has_moved_text();Hypocoercivity

Cédric Villani, Unité de Mathématiques Pures et Appliquées, UMR CNRS 5669, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, F-69364 Lyon Cedex 07, FRANCE

Publication: Memoirs of the American Mathematical Society
Publication Year: 2009; Volume 202, Number 950
ISBNs: 978-0-8218-4498-4 (print); 978-1-4704-0564-9 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-09-00567-5
Published electronically: July 22, 2009
MathSciNet review: 2562709
Keywords:Convergence to equilibrium; hypoellipticity; hypocoercivity; Fokker–Planck equation; Boltzmann equation
MSC: Primary 35Q84; Secondary 35H10, 76N10, 76P05, 82C70

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Part I. $L = A^*A + B$

• 1. Notation
• 2. Operators $L = A^*A + B$
• 3. Coercivity and hypocoercivity
• 4. Basic theorem
• 5. Generalization
• 6. Hypocoercivity in entropic sense
• 7. Application: The kinetic Fokker-Planck equation
• 8. The method of multipliers
• 9. Further applications and open problems

Part II. The auxiliary operator method

• 10. Assumptions
• 11. Main theorem
• 12. Simplified theorem and applications
• 13. Discussion and open problems

Part III. Fully nonlinear equations

• 14. Main abstract theorem
• 15. Proof of the main theorem
• 16. Compressible Navier-Stokes system
• 17. Weakly self-consistent Vlasov-Fokker-Planck equation
• 18. Boltzmann equation

Appendices

• A.19. Some criteria for Poincaré inequalities
• A.20. Well-posedness for the Fokker-Planck equation
• A.21. Some methods for global hypoellipticity
• A.22. Local positivity estimates
• A.23. Toolbox

### Abstract

This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, taking the general form ${\frac{\partial f}{\partial t}}+ L f =0$. The question is whether and how one can overcome the degeneracy by exploiting commutators.

In Part I, the focus is on a class of operators taking the abstract form $L= A^*A+B$ in a Hilbert space. A general Hilbertian result is proven, which can be considered as a spectral'' counterpart of Hörmander's regularity theorem. Then I discuss an entropic'' version of this result, which leads to more general statements but needs more structure. The main example of application is the linear Fokker-Planck equation; other examples are discussed.

In Part II, a different method is discussed, based on the introduction of an auxiliary operator which has good commutation and non-commutation properties with $L$. Some recent results are reinterpreted in this formalism.

In Part III, a third method is discussed, applying to nonlinear equations with very little structure. This one is the most general but needs a lot of smoothness, and does not in general achieve the exponential convergence. Applications to various models of fluid mechanics, in particular the Boltzmann equation, are discussed. My recent results with Desvillettes about the convergence to equilibrium for the Boltzmann equation are extended and simplified in this way.

The unity of the three parts comes from the method: in all cases, the convergence to equilibrium is obtained by a carefully designed Lyapunov functional, or family of Lyapunov functionals. Many open problems and possible directions for future research are discussed throughout the text.

In a long Appendix, I introduce some methods for the study of global hypoellipticity, focusing on the kinetic Fokker-Planck equation once again.