Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations I. Non-cutoff case and Maxwellian molecules

Mathematical Models and Methods in Applied Sciences

Volumn 15, Issue 6, 2005, Pages 907-920

  Alexandre, Radjesvarane ^a   El Safadi, Mouhamad ^a  

^a UNIVERSITÉ D'ORLÉANS   (France)

Author keywords

Boltzmann equation; Littlewood Paley theory; Maxwellian molecules; Noncutoff collision cross sections

Indexed keywords

EID: 18844411595     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202505000613     Document Type: Article

References (23)

1. R. Alexandre, Around 3D Boltzmann nonlinear operator without cutoff. A new formulation, Math. Mod. Appl. Numer. 34-3 (2000) 575-590.
2. R. Alexandre, Solutions of Boltzmann equation without cutoff and for small initial data, J. Stat. Phys. 104 (2002) 327-358.
3. R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long range interactions, Arch. Rat. Mech. Anal. 152-4 (2000) 327-355.
4. L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rat. Mech. Anal. 77 (1981) 11-21.
5. A.-V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules, Math. Phys. Rev. 7 (1988) 111-233.
6. C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, 1988).
7. S. Chapman and T. Cowling, The Mathematical Theory of Non Uniform Gases, (Cambridge Univ. Press, 1952).
8. L. Desvillettes, About the regularization properties of the non cut-off Kac equation, Comm. Math. Phys. 168 (1995) 417-440.
9. L. Desvillettes, Regularization properties of the 2-dimensional non radially symmetric non cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Trans. Theory Stat. Phys. 26-3 (1997) 341-357.
10. L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. P.D.E 29 (2004) 133-155.
11. N. Fournier, Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach, Ann. Appl. Probab. 10 (2000) 434-462.
12. N. Fournier and S. Méléard, Existence results for 2D homogeneous Boltzmann equation without cutoff and for non Maxwell molecules by use of Malliavin calculus, preprint 622 (2000) Université Paris VI.
13. T. Goudon, On Boltzmann equation and Fokker-Planck asymptotics: influence of grazing collisions, J. Statist. Phys. 89 (1997) 751-776.
14. C. Graham and S. Méléard, Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations, Comm. Math. Phys. 205 (1999) 551-569.
15. P.-G. Lemarié, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Vol. 431 (Chapman and Hall/CRC, 2002).
16. A. Proutière, A new results of regularization for weak solutions of Boltzmann equation, preprint (1996), unpublished.
17. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear PdE (De Gruyter, 1996).
18. L. Schwartz, Théorie des Distributions (Hermann, 1966).
19. E.-M. Stein, Harmonic Analysis. Real Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, 1993).
20. H. Triebel, Theory of Function Spaces (Birkhauser, 1983).
21. T. Tao, Littlewood-Paley Decomposition, Lecture Notes 254A, See website http://www.math.ucla.edu/~tao/
22. C. Villani, On a new class of solutions for the spatially homogeneous, Boltzmann and Landau equations, Arch. Rat. Mech. Anal. 143 (1998) 273-307.
23. C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Handbook of Fluid Mechanics, eds. S. Friedlander and D. Serre (North-Holland, 2002).