Uncertainty principle and kinetic equations

Journal of Functional Analysis

Volumn 255, Issue 8, 2008, Pages 2013-2066

  Alexandre, R ^a   Morimoto, Y ^b   Ukai, S ^c   Xu, C J ^d   Yang, T ^c  

^a IRENav   (France)

^b KYOTO UNIVERSITY   (Japan)

^c CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^d UNIVERSITÉ DE ROUEN   (France)

Author keywords

Boltzmann equations; Kinetic equations; Landau equation; Microlocal analysis; Non cutoff cross sections; Uncertainty principle

Indexed keywords

EID: 56349126806     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2008.07.004     Document Type: Article

References (31)

1. Alexandre R. On some related non-homogeneous 3D Boltzmann models in the non-cutoff cas. J. Math. Kyoto Univ. 40 3 (2000) 493-524
2. Alexandre R. Integral estimates for linear singular operator linked with Boltzmann operator. Part I: Small singularities 0 < ν < 1. Indiana Univ. Math. J. 55 6 (2006) 1975-2021
3. Alexandre R. Lecture notes on Boltzmann equation and singular cross sections, based on talks given in Shanghai. available via. http://wwwold.ecole-navale.fr/fr/irenav/cv/radjesvarane.alexandre/versio n-26-04-07.pdf
4. Alexandre R., and Safadi M. Littlewood-Paley decomposition and regularity issues in Boltzmann equation homogeneous equations. I. Non-cutoff and Maxwell cases. Math. Meth. Model. Appl. Sci. (2005) 8-15
5. Alexandre R., and Villani C. On the Boltzmann equation for long-range interaction. Comm. Pure Appl. Math. 55 (2002) 30-70
6. Alexandre R., Desvillettes L., Villani C., and Wennberg B. Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 (2000) 327-355
7. R. Alexandre, Y. Morimoto, S. Ukai, C.-Y. Xu, T. Yang, Full regularity for the non-homogeneous Boltzmann equation without cutoff, in preparation
8. Bobylev A. The theory of nonlinear, spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C Math. Phys. 7 (1988) 111-233
9. Bouchut F. Hypoelliptic regularity in kinetic equations. J. Math. Pure Appl. 81 (2002) 1135-1159
10. Bouchut F., Golse F., and Pulvirenti M. Kinetic Equations and Asymptotic Theory. Ser. Appl. Math. (Paris) (2000), Gauthiers-Villars, Paris
11. Y. Chen, L. Desvillettes, L. He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal., in press
12. Desvillettes L. About the use of the Fourier transform for the Boltzmann equation. Summer School on Methods and Models of Kinetic Theory, M&MKT, 2002. Riv. Mat. Univ. Parma (7) 2 (2003) 1-99
13. Desvillettes L., and Villani C. On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25 1-2 (2000) 179-259
14. 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
15. L. Desvillettes, G. Furiolo, E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Trans. Amer. Math. Soc., in press
16. Fefferman C. The uncertainty principle. Bull. Amer. Math. Soc. 9 (1983) 129-206
17. Folland G.B., and Sitaram A. The uncertainty principle: A mathematical survey. J. Fourier Anal. Appl. 3 3 (1997) 207-238
18. Grad H. Asymptotic theory of the Boltzmann equation II. In: Laurmann J.A. (Ed). Rarefied Gas Dynamics, vol. 1 (1963), Academic Press, New York 26-59
19. Guo Y. The Landau equation in a periodic box. Comm. Math. Phys. 231 3 (2002) 391-434
20. Guo Y. The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55 9 (2002) 1104-1135
21. Kato T. Perturbation Theory for Linear Operators (1995), Springer-Verlag, Berlin
22. Morimoto Y. The uncertainty principle and hypoelliptic operators. Publ. Res. Inst. Math. Sci. 23 (1987) 955-964
23. Morimoto Y. Estimates for degenerate Schrödinger operators and hypoellipticity for infinitely degenerate elliptic operators. J. Math. Kyoto Univ. 32 (1992) 333-372
24. Morimoto Y., and Morioka T. The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators. Bull. Sci. Math. 121 (1997) 507-547
25. Morimoto Y., and Xu C.-J. Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators. Astérisque 284 (2003) 245-264
26. Morimoto Y., and Xu C.-J. Hypoelliticity for a class of kinetic equations. J. Math. Kyoto Univ. 47 (2007) 129-152
27. Morimoto Y., and Xu C.-J. Nonlinear hypoellipticity of infinite type. Funkcial. Ekvac. 50 (2007) 33-65
28. Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst., in press
29. Ukai S. Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan J. Appl. Math. 1 1 (1984) 141-156
30. 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
31. Villani C. A review of mathematical topics in collisional kinetic theory. In: Friedlander S., and Serre D. (Eds). Handbook of Mathematical Fluid Dynamics, vol. 1 (2002), North-Holland, Amsterdam 71-305