Integral estimates for a linear singular operator linked with the Boltzmann operator. Part I: Small singularities 0 < v < 1

Indiana University Mathematics Journal

Volumn 55, Issue 6, 2006, Pages 1975-2021

  Alexandre, Radjesvarane ^a  

^a UNIVERSITÉ D EVRY VAL D ESSONNE   (France)

Author keywords

Boltzmann equation; Littlewood Paley theory; Singular collision cross sections

Indexed keywords

EID: 33846861108     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2006.55.2861     Document Type: Article

References (40)

1. RADJESVARANE ALEXANDRE, Around 3D Boltzmann non linear operator without angular cutoff. A new formulation, M2AN Math. Model. Numer. Anal. 34 (2000), 575-590, http://dx.doi.org/10.1051/m2an:2000157. MR 1763526 (2002a:82052)
2. _, Remarks on 3D Boltzmann linear equation without cutoff, Transport Theory Statist. Phys. 28 (1999), 433-473. MR 1705619 (2001c:82031)
3. RADJESVARANE ALEXANDRE, LAURENT DESVILLETTES, CÉDRIC VILLANI, and BERNT WENNBERG, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000), 327-355, http://dx.doi.org/10.1007/s002050000083. MR 1765272 (2001c:82061)
4. RADJESVARANE ALEXANDRE and MOUHAMAD EL SAFADI, Littlewood Paley decomposition and regularity issues in Boltzmann homogeneous equations. I. Non cutoff and Maxwell cases, M3AS (2005), 8-15.
5. _, Littlewood Paley decomposition and regularity issues in Boltzmann homogeneous equations. II. Non cutoff and non-Maxwell cases, work in progress.
6. RADJESVARANE ALEXANDRE and CÉDRIC VILLANI, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math. 55 (2002), 30-70, http://dx.doi.org/10.1002/cpa. 10012. MR 1857879 (2002f:82026)
7. LUISA ARLOTTI, NICOLA BELLOMO, MIROSŁAW LACHOWICZ, JACEK POLEWCZAK, and WŁODZIMIERZ WALUŚ, Lecture Notes on the Mathematical Theory of the Boltzmann equation. Series on Advances in Mathematics for Applied Sciences, edited by Nicola Bellomo, vol. 33, World Scientific Publishing Co. Inc., River Edge, NJ, 1995, ISBN 981-02-2166-5. MR 1459984 (98g: 82036)
8. JÖRAN BERUH and JÖRGEN LÖFSTRÖM, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976. MR 0482275 (58 #2349)
9. HUY QUI BUI, Weighted Besov and Triebel spaces: Interpolation by the real method, Hiroshima Math.J. 12 (1982), 581-605. MR 676560 (84f:46038)
10. CARLO CERCIGNANI, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988, ISBN 0-387-96637-4. MR 1313028 (951:82082)
11. CARLO CERCIGNANI, REINHARD ILLNER, and MARIO PULVIRENTI, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994, ISBN 0-387-94294-7. MR 1307620 (96g:82046)
12. SYDNEY CHAPMAN and T. G. COWLING, The mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, Cambridge University Press, New York, 1960. MR 0116537 (22 #7324)
13. RONALD R. COIFMAN and YVES MEYER, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978. MR 518170 (81b:47061) (French)
14. ∞, weights and applications, J. Funct. Anal. 213 (2004), 412-439, http://dx.doi.org/10.1016/j.jfa.2003. 09.002. MR 2078632 (2005g:42029)
15. GUILLERMO P CURBERA, JOSÉ GARCÍA-CUERVA, JOSÉ MARIA MARTELL, and CARLOS PÉREZ, Extrapolation with weights. Rearrangement invariant function spaces. Modular inequalities and Applications to singular integrals, Adv. Math, (to appear).
16. LAURENT DESVILLETTES, Regularization properties of the 2-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Transport Theory Statist. Phys. 26 (1997), 341-357. MR 1475459 (98h:82058)
17. LAURENT DESVILLETTES and BERNT WENNBERG, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations 29 (2004), 133-155, http://dx.doi.org/10.1081/PDE-120028847. MR 2038147 (2004k:82086)
18. RONALD J. DIPERNA and PIERRE-LOUIS LIONS, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), 321-366, http://dx.doi.org/10.2307/1971423. MR 1014927 (90k:82045)
19. JAVIER DUOANDIKOETXEA, Fourier Analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001, ISBN 0-8218-2172-5. MR 1800316 (2001k:42001)
20. JOSÉ GARCÍA-CUERVA and JOSÉ L. RUBIO DE FRANCIA, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985, ISBN 0-444-87804-1. MR 807149 (87d:42023)
21. ROBERT T. GLASSEY, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996, ISBN 0-89871-367-6. MR 1379589 (97i:82070)
22. PIERRE GILLES LEMARIÉ- RIEUSSET, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002, ISBN 1-58488-220-4. MR 1938147 (2004e:35178)
23. PIERRE-LOUIS LIONS, Compactness in Boltzmann's equation via Fourier integral operators and applications. II, J. Math. Kyoto Univ. 34 (1994), 429-461. MR 1284432 (96B5133)
24. _, Compactness in Boltzmann's equation via Fourier integral operators and applications. III, J. Math. Kyoto Univ. 34 (1994), 539-584. MR 1295942 (96f:35134)
25. CLÉMENT MOUHOT, Explicit coercivity estimates for the linearized Boltzmann and Landau operators (preprint), available through http://www.hyke.org.
26. CLÉMENT MOUHOT and CÉDRIC VILLANI, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal. 173 (2004), 169-212, http://dx.doi.org/10.1007/s00205-004-0316-7. MR 2081030 (2006f:82073)
27. THOMAS RUNST and WINFRIED SICKEL, Sobolev Spaces of 'Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996, ISBN 3-11-015113-8. MR 1419319 (98a:47071)
28. locp weights, Math. Nachr. 224 (2001), 145-180, http://dx.doi.org/10.1002/1522-2616(200104)224:1;145::AID- MANA145¿3.0.CO;2-2. MR 1821243 (2002k:42045)
29. MOUHAMAD EL SAFADI, Littlewood-Paley theory and homogeneous Landau equation (preprint; submitted).
30. ELIAS M. STEIN, Harmonic Analysis: Rreal-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, ISBN 0-691-03216-5. MR 1232192 (95c:42002)
31. [31 ] LUC TARTAR, An Introduction to Sobolev spaces and Interpolation spaces. Lectures Notes, available through http://www.math.cmu.edu/cna/publications.html.
32. MICHAEL E. TAYLOR, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston Inc., Boston, MA, 1991, ISBN 0-8176-3595-5. MR 1121019 (92j:35193)
33. _, Partial Differential Equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996, ISBN 0-387-94653-5. MR 1395148 (98b:35002b)
34. _, Partial Differential Equations. II, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996, ISBN 0-387-94651-9. MR 1395149 (98b:35003)
35. _, Partial Differential Equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997, ISBN 0-387-94652-7. MR 1477408 (98k:35001)
36. HANS TRIEBEL, Theory of Function Spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983, ISBN 3-7643-1381-1. MR 781540 (86j:46026)
37. _, Theory of Function Spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992, ISBN 3-7643-2639-5. MR 1163193 (93f:46029)
38. CÉDRIC VILLANI, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, Vol. I (S. Friedlander and D. Serre, eds.), North-Holland, Amsterdam, 2002, pp. 71-305. MR 1942465 (2003k:82087)
39. _, Cours Peccot, Collège de France, 2003, unpublished notes.
40. BERNT WENNBERG, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations 19 (1994), 2057-2074. MR 1301182 (95k:76103)