Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations

Journal of Differential Equations

Volumn 246, Issue 1, 2009, Pages 320-339

  Chen, Hua ^a   Li, Wei Xi ^a   Xu, Chao Jiang ^a,b  

^a WUHAN UNIVERSITY   (China)

^b UNIVERSITÉ DE ROUEN   (France)

Author keywords

Fokker Planck equation; Gevrey regularity; Subelliptic estimate

Indexed keywords

EID: 55249097148     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.05.019     Document Type: Article

References (21)

1. Alexandre R., Ukai S., Morimoto Y., Xu C.-J., and Yang T. Uncertainty principle and regularity for Boltzmann equation. C. R. Math. Acad. Sci. Paris 345 12 (2007) 673-677
2. Alexandre R., Desvillettes L., Villani C., and Wennberg B. Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 (2000) 327-355
3. Alexandre R., and Safadi M. Littlewood-Paley decomposition and regularity issues in Boltzmann homogeneous equations. I. Non cutoff and Maxwell cases. M3AS 15 (2005) 907-920
4. Bouchut F. Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. 81 (2002) 1135-1159
5. Chen H., and Rodino L. General theory of PDE and Gevrey class. General Theory of Partial Differential Equations and Microlocal Analysis. Trieste, 1995. Pitman Res. Notes Math. Ser. vol. 349 (1996), Longman, Harlow 6-81
6. Chen H., and Rodino L. Paradifferential calculus in Gevrey class. J. Math. Kyoto Univ. 41 (2001) 1-31
7. Derridj M., and Zuily C. Sur la régularité Gevrey des opérateurs de Hörmander. J. Math. Pures Appl. 52 (1973) 309-336
8. Desvillettes L., and Wennberg B. Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm. Partial Differential Equations 29 1-2 (2004) 133-155
9. Durand M. Régularité Gevrey d'une classe d'opérateurs hypo-elliptiques. J. Math. Pures Appl. 57 (1978) 323-360
10. Friedman A. On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations. J. Math. Mech. 7 (1958) 43-59
11. Helffer B., and Nier F. Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Lecture Notes in Math. vol. 1862 (2005), Springer-Verlag, Berlin
12. 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
13. Hörmander L. Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147-171
14. Morimoto Y., and Xu C.-J. Hypoellipticity for a class of kinetic equations. J. Math. Kyoto Univ. 47 1 (2007) 129-152
15. Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, preprint
16. Rodino L. Linear Partial Differential Operators in Gevrey Class (1993), World Scientific, Singapore
17. Rothschild L.P., and Stein E.M. Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1977) 248-315
18. Treves F. Introduction to Pseudodifferential and Fourier Integral Operators (1980), Plenum, New York
19. Ukai S. Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan J. Appl. Math. 1 1 (1984) 141-156
20. Villani C. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143 (1998) 273-307
21. Xu C.-J. Nonlinear microlocal analysis. General Theory of Partial Differential Equations and Microlocal Analysis. Trieste, 1995. Pitman Res. Notes Math. Ser. vol. 349 (1996), Longman, Harlow 155-182