Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics

Journal of Differential Equations

Volumn 247, Issue 8, 2009, Pages 2209-2234

  Carrillo, José A ^a,b   Laurençot, Philippe ^c   Rosado, Jesús ^b  

^a INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS ICREA   (Spain)

^b UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^c UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 68949158435     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.07.018     Document Type: Article

References (28)

1. Burger M., di Francesco M., and Dolak Y. The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38 (2006) 1288-1315
2. Carrillo J.A., Jüngel A., Markowich P., Toscani G., and Unterreiter A. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133 (2001) 1-82
3. Carrillo J.A., McCann R.J., and Villani C. Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179 (2006) 217-263
4. Carrillo J.A., and Toscani G. Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations. Math. Methods Appl. Sci. 21 (1998) 1269-1286
5. Carrillo J.A., Rosado J., and Salvarani F. 1D nonlinear Fokker-Planck equations for fermions and bosons. Appl. Math. Lett. 21 (2008) 148-154
6. Chavanis P.-H. Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics and two-dimensional turbulence. Phys. Rev. E 68 (2003) 036108
7. Chavanis P.-H. Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski. Phys. A 332 (2004) 89-122
8. Chavanis P.-H. Generalized Fokker-Planck equations and effective thermodynamics. Phys. A 340 (2004) 57-65
9. Dellacherie C., and Meyer P.A. Probabilités et Potentiel (1975), Hermann, Paris
10. Dolbeault J. Kinetic models and quantum effects: A modified Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal. 127 (1994) 101-131
11. N. J. Funct. Anal. 100 (1991) 119-161
12. Frank T.D. Classical Langevin equations for the free electron gas and blackbody radiation. J. Phys. A 37 (2004) 3561-3567
13. Frank T.D. Nonlinear Fokker-Planck Equations. Springer Ser. Synergetics (2005), Springer
14. 2. Arch. Ration. Mech. Anal. 163 (2002) 209-258
15. Kaniadakis G. Generalized Boltzmann equation describing the dynamics of bosons and fermions. Phys. Lett. A 203 (1995) 229-234
16. Kaniadakis G. H-theorem and generalized entropies within the framework of nonlinear kinetics. Phys. Lett. A 288 (2001) 283-291
17. Kaniadakis G., Lapenta G., and Quarati P. Stochastic evolution of systems of particles obeying an exclusion principle. Phys. A 225 (1996) 323-335
18. Kaniadakis G., and Quarati P. Kinetic equation for classical particles obeying an exclusion principle. Phys. Rev. E 48 (1993) 4263-4270
19. Ladyženskaja O.A., Solonnikov V.A., and Ural'ceva N.N. Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monogr. vol. 23 (1968), Amer. Math. Soc., Providence, RI
20. ème cycle, Université de Paris VI, 1977
21. Lemarié-Rieusset P.G. Recent Developments in the Navier-Stokes Problem (2002), Chapman & Hall-CRC, Boca-Raton
22. Lu X., and Wennberg B. On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal. 168 (2003) 1-34
23. Mouhot C., and Neumann L. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19 (2006) 969-998
24. Neumann L., and Schmeiser C. Convergence to global equilibrium for a kinetic model for fermions. SIAM J. Math. Anal. 36 (2005) 1652-1663
25. Neumann L., and Sparber C. Stability of steady states in kinetic Fokker-Planck equations for bosons and fermions. Commun. Math. Sci. 5 (2007) 765-777
26. Rossani A., and Kaniadakis G. A generalized quasi-classical Boltzmann equation. Phys. A 277 (2000) 349-358
27. Toscani G. Remarks on entropy and equilibrium states. Appl. Math. Lett. 12 (1999) 19-25
28. Vázquez J.L. The Porous Medium Equation: Mathematical Theory. Oxford Math. Monogr. (2007), Clarendon Press/Oxford Univ. Press, Oxford