`a`
Kinetic and Related Models (KRM)
 

On a relativistic Fokker-Planck equation in kinetic theory
Pages: 401 - 426, Volume 4, Issue 2, June 2011

doi:10.3934/krm.2011.4.401      Abstract        References        Full text (557.6K)           Related Articles

José Antonio Alcántara - Departamento de Matemática Aplicada, Facultad de ciencias, Universidad de Granada, 18071 Granada, Spain (email)
Simone Calogero - Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (email)

1 F. Andreu, V. Caselles, J. M. Mazón and S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal., 182 (2006), 269-297.       
2 M. Bostan and T. Goudon, Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case, Kinet. Relat. Models, 1 (2008), 139-170.       
3 F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with coulombic and Newtonian potentials, Diff. Integ. Eqs., 8 (1995), 487-514.       
4 S. Calogero, Spherical symmetric steady states of galactic dynamics in scalar gravity, Class. Quant. Grav., 20 (2003), 1729-1741.       
5 S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system, Comm. Math. Phys., 266 (2006), 343-353.       
6 S. Calogero, O. Sánchez and J. Soler, Asymptotic behavior and orbital stability of galactic dynamics in relativistic scalar gravity, Arch. Rat. Mech. Anal., 194 (2009), 743-773.       
7 S. Calogero, J. Calvo, O. Sánchez and J. Soler, Virial inequalities for steady states in relativistic galactic dynamics, Nonlinearity, 23 (2010), 1851-1871.       
8 S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations on Riemannian manifolds, preprint, arXiv:1009.5086.
9 J. A. Carrillo, P. Laurençot and J. Rosado, Fermi-Dirac-Fokker-Planck equation: Well-posedness $&$ long-time asymptotics, J. Diff. Eqns., 247 (2009), 2209-2234.       
10 G. Chac\'on-Acosta and G. M. Kramer, Fokker-Planck-type equations for a simple gas and for a semirelativistic Brownian motion from a relativistic kinetic theory, Phys. Rev. E., 76 (2007), 021201.
11 S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev Mod. Phys., 15 (1943), 1-89.       
12 F. Debbasch and C. Chevalier, Relativistic stochastic processes: A review, AIP Conf. Proc., 913 (2007), 42-48.       
13 J. Dolbeault, Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: External potential and confinement (large time behavior and steady states), J. Math. Pures Appl., 78 (1999), 121-157.       
14 K. Dressler, Steady states in plasma physics-the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci., 12 (1990), 471-487.       
15 K. Dressler, Stationary solutions of the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci., 9 (1987), 169-176.       
16 J. Dunkel and P. Hänggi, Theory of the relativistic Brownian motion: The (1+3)-dimensional case, Phys. Rev. E, 72 (2005), 036106.       
17 J. Dunkel and P. Hänggi, Relativistic Brownian motion, Phys. Rep., 471 (2009), 1-73.       
18 D. T. Frank, "Nonlinear Fokker-Planck Equations: Fundamentals and Applications," Springer Series in Synergetics 25, Springer-Verlag, New York, 2005.
19 R. T. Glassey, J. Schaeffer and Y. Zheng, Steady states of the Vlasov-Poisson-Fokker-Planck system, J. Math. An. Appl., 202 (1996), 1058-1075.       
20 Z. Haba, Relativistic diffusion, Phys. Rev. E, 79 (2009), 021128.
21 Z. Haba, Relativistic diffusion of elementary particles with spin, Journ. Phys. A, 42 (2009), 445401.       
22 Z. Haba, Energy and entropy of relativistic diffusing particles, Mod. Phys. Lett. A, 25 (2010), 2683-2695.       
23 Z. Haba, Relativistic diffusive transport, preprint arXiv:0911.3126.
24 B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators And Witten Laplacians," Lecture Notes in Mathematics 1862, Springer-Verlag, New York, 2000.       
25 L. Hörmander, Pseudodifferential operators and non-elliptic boundary problems, Ann. of Math., 83 (1966), 129-209.       
26 F. John, Blow-up for quasi linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.       
27 I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Graduate Texts in Mathematics 113 2nd edition, Springer-Verlag, New York, 1991.       
28 C. R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Math. Meth. Appl. Sci., 21 (1998), 1287-1296.
29 E. H. Lieb and M. Loss, "Analysis," American Math. Soc. 14, Providence, 1997.       
30 C.-P. Ma and E. Bertschinger, A cosmological kinetic theory for the evolution of cold dark matter halos with substructure: Quasi-linear theory, The Astroph. J., 612 (2004), 28-49.
31 M. Risken, "The Fokker-Planck Equation: Methods of Solutions and Applications," Springer Series in Synergetics 18, Springer-Verlag, Berlin, 1996.       
32 M. Schunck, M. Hegmann and E. Sedlmayr, The influence of stochastic density fluctuations on the infrared emissions of interstellar dark clouds, Mon. Noti. Royal Astron. Soc., 374 (2007), 949-959.
33 S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity, Phys. Rev. D, 47 (1993), 1529-1540.
34 C. Sogge, "Lectures on Nonlinear Wave Equations," International Press, Cambridge, 1995.       
35 J. L. Vázquez, "The Porous Medium Equation: Mathematical Theory," Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, Oxford 2007.
36 T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488.       
37 C. Villani Hypocoercivity, Memoirs of the AMS, 202 (2009), n. 950.

Go to top