Distributional and classical solutions to the cauchy boltzmann problem for soft potentials with integrable angular cross section

Journal of Statistical Physics

Volumn 137, Issue 5, 2009, Pages 1147-1165

  Alonso, Ricardo J ^a   Gamba, Irene M ^b  

^a Rice University   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Boltzmann equation for soft potentials; Generalized and classical solutions; Stability in Lp spaces

Indexed keywords

EID: 74449088828     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-009-9873-3     Document Type: Article

References (26)

1. Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55, 30-70 (2002).
2. Alonso, R.: Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data. Indiana Univ. Math. J. 58, 999-1022 (2009).
3. Alonso, R., Carneiro, E.: Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform. Adv. Math. (to appear).
4. Alonso, R., Carneiro, E., Gamba, I.: Convolution inequalities for the Boltzmann collision operator. arXiv: 0902. 0507 [math. AP] (submitted for publication).
5. Bellomo, N., Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behavior. J. Math. Phys. 26, 334-338 (1985).
6. Boudin, L., Desvillettes, L.: On the singularities of the global small solutions of the full Boltzmann equation. Monatsh. Math. 131, 91-108 (2000).
7. Caflisch, R.: The Boltzmann equation with a soft potential (II). Commun. Math. Phys. 74, 97-109 (1980).
8. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Appl. Math. Sci. Springer, Berlin (1994).
9. Diperna, R., Lions, P.-L.: On the Cauchy problem for the Boltzmann equation. Ann. Math. 130, 321-366 (1989).
10. Glassey, R.: Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data. Commun. Math. Phys. 264, 705-724 (2006).
11. Goudon, T.: Generalized invariant sets for the Boltzmann equation. Math. Models Methods Appl. Sci. 7, 457-476 (1997).
12. Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169, 305-353 (2003).
13. Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153, 593-630 (2003).
14. Ha, S.-Y.: Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Differ. Equ. 215, 178-205 (2005).
15. Ha, S.-Y., Yun, S.-B.: Uniform L1-stability estimate of the Boltzmann equation near a local Maxwellian. Phys. Nonlinear Phenom. 220, 79-97 (2006).
16. Hamdache, K.: Existence in the large and asymptotic behavior for the Boltzmann equation. Jpn. J. Appl. Math. 2, 1-15 (1985).
17. Hamdache, K.: Initial boundary value problems for Boltzmann equation. Global existence of week solutions. Arch. Ration. Mech. Anal. 119, 309-353 (1992).
18. Illner, R., Shinbrot, M.: The Boltzmann equation, global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95, 217-226 (1984).
19. Kaniel, S., Shinbrot, M.: The Boltzmann equation I. Uniqueness and local existence. Commun. Math. Phys. 58, 65-84 (1978).
20. Mischler, S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210, 447-466 (2000).
21. Mischler, S., Perthame, B.: Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to Maxwellian. SIAM J. Math. Anal. 28, 1015-1027 (1997).
22. Palczewski, A., Toscani, G.: Global solution of the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian. J. Math. Phys. 30, 2445-2450 (1989).
23. Toscani, G.: On the nonlinear Boltzmann equation in unbounded domains. Arch. Ration. Mech. Anal. 95, 37-49 (1986).
24. Toscani, G.: Global solution of the initial value problem for the Boltzmann equation near a local Maxwellian. Arch. Ration. Mech. Anal. 102, 231-241 (1988).
25. Ukai, S., Asano, K.: On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18, 477-519 (1982) (57-99).
26. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143, 273-307 (1998).