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On a relativistic Fokker-Planck equation in kinetic theory
1. | Departamento de Matemática Aplicada, Facultad de ciencias, Universidad de Granada, 18071 Granada, Spain |
2. | Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada |
References:
[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.
doi: 10.1007/s00205-006-0428-3. |
[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.
|
[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.
|
[4] |
S. Calogero, Spherical symmetric steady states of galactic dynamics in scalar gravity,, Class. Quant. Grav., 20 (2003), 1729.
doi: 10.1088/0264-9381/20/9/310. |
[5] |
S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system,, Comm. Math. Phys., 266 (2006), 343.
doi: 10.1007/s00220-006-0029-x. |
[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.
doi: 10.1007/s00205-008-0173-x. |
[7] |
S. Calogero, J. Calvo, O. Sánchez and J. Soler, Virial inequalities for steady states in relativistic galactic dynamics,, Nonlinearity, 23 (2010), 1851.
doi: 10.1088/0951-7715/23/8/004. |
[8] |
S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations on Riemannian manifolds,, preprint, (). Google Scholar |
[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.
doi: 10.1016/j.jde.2009.07.018. |
[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).
doi: 10.1103/PhysRevE.76.021201. |
[11] |
S. Chandrasekhar, Stochastic problems in physics and astronomy,, Rev Mod. Phys., 15 (1943), 1.
doi: 10.1103/RevModPhys.15.1. |
[12] |
F. Debbasch and C. Chevalier, Relativistic stochastic processes: A review,, AIP Conf. Proc., 913 (2007), 42.
doi: 10.1063/1.2746722. |
[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.
doi: 10.1016/S0021-7824(01)80006-4. |
[14] |
K. Dressler, Steady states in plasma physics-the Vlasov-Fokker-Planck equation,, Math. Meth. Appl. Sci., 12 (1990), 471.
doi: 10.1002/mma.1670120603. |
[15] |
K. Dressler, Stationary solutions of the Vlasov-Fokker-Planck equation,, Math. Meth. Appl. Sci., 9 (1987), 169.
doi: 10.1002/mma.1670090113. |
[16] |
J. Dunkel and P. Hänggi, Theory of the relativistic Brownian motion: The (1+3)-dimensional case,, Phys. Rev. E, 72 (2005).
doi: 10.1103/PhysRevE.72.036106. |
[17] |
J. Dunkel and P. Hänggi, Relativistic Brownian motion,, Phys. Rep., 471 (2009), 1.
doi: 10.1016/j.physrep.2008.12.001. |
[18] |
D. T. Frank, "Nonlinear Fokker-Planck Equations: Fundamentals and Applications,", Springer Series in Synergetics {\bf 25}, 25 (2005). Google Scholar |
[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.
doi: 10.1006/jmaa.1996.0360. |
[20] |
Z. Haba, Relativistic diffusion,, Phys. Rev. E, 79 (2009).
doi: 10.1103/PhysRevE.79.021128. |
[21] |
Z. Haba, Relativistic diffusion of elementary particles with spin,, Journ. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/44/445401. |
[22] |
Z. Haba, Energy and entropy of relativistic diffusing particles,, Mod. Phys. Lett. A, 25 (2010), 2683.
doi: 10.1142/S0217732310033992. |
[23] |
Z. Haba, Relativistic diffusive transport,, preprint \arXiv{0911.3126}., (). Google Scholar |
[24] |
B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators And Witten Laplacians,", Lecture Notes in Mathematics {\bf 1862}, 1862 (1862).
|
[25] |
L. Hörmander, Pseudodifferential operators and non-elliptic boundary problems,, Ann. of Math., 83 (1966), 129.
doi: 10.2307/1970473. |
[26] |
F. John, Blow-up for quasi linear wave equations in three space dimensions,, Comm. Pure Appl. Math., 34 (1981), 29.
doi: 10.1002/cpa.3160340103. |
[27] |
I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", Graduate Texts in Mathematics {\bf 113} 2$^{nd}$ edition, 113 (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.
doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G. |
[29] |
E. H. Lieb and M. Loss, "Analysis,", American Math. Soc. {\bf 14}, 14 (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. Google Scholar |
[31] |
M. Risken, "The Fokker-Planck Equation: Methods of Solutions and Applications,", Springer Series in Synergetics {\bf 18}, 18 (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.
doi: 10.1111/j.1365-2966.2006.11215.x. |
[33] |
S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity,, Phys. Rev. D, 47 (1993), 1529.
doi: 10.1103/PhysRevD.47.1529. |
[34] |
C. Sogge, "Lectures on Nonlinear Wave Equations,", International Press, (1995).
|
[35] |
J. L. Vázquez, "The Porous Medium Equation: Mathematical Theory,", Oxford Math. Monogr., (2007). Google Scholar |
[36] |
T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system,, SIAM J. Math. Anal., 42 (2010), 459.
doi: 10.1137/090755796. |
[37] |
C. Villani, Hypocoercivity,, Memoirs of the AMS, 202 (2009). Google Scholar |
show all references
References:
[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.
doi: 10.1007/s00205-006-0428-3. |
[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.
|
[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.
|
[4] |
S. Calogero, Spherical symmetric steady states of galactic dynamics in scalar gravity,, Class. Quant. Grav., 20 (2003), 1729.
doi: 10.1088/0264-9381/20/9/310. |
[5] |
S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system,, Comm. Math. Phys., 266 (2006), 343.
doi: 10.1007/s00220-006-0029-x. |
[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.
doi: 10.1007/s00205-008-0173-x. |
[7] |
S. Calogero, J. Calvo, O. Sánchez and J. Soler, Virial inequalities for steady states in relativistic galactic dynamics,, Nonlinearity, 23 (2010), 1851.
doi: 10.1088/0951-7715/23/8/004. |
[8] |
S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations on Riemannian manifolds,, preprint, (). Google Scholar |
[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.
doi: 10.1016/j.jde.2009.07.018. |
[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).
doi: 10.1103/PhysRevE.76.021201. |
[11] |
S. Chandrasekhar, Stochastic problems in physics and astronomy,, Rev Mod. Phys., 15 (1943), 1.
doi: 10.1103/RevModPhys.15.1. |
[12] |
F. Debbasch and C. Chevalier, Relativistic stochastic processes: A review,, AIP Conf. Proc., 913 (2007), 42.
doi: 10.1063/1.2746722. |
[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.
doi: 10.1016/S0021-7824(01)80006-4. |
[14] |
K. Dressler, Steady states in plasma physics-the Vlasov-Fokker-Planck equation,, Math. Meth. Appl. Sci., 12 (1990), 471.
doi: 10.1002/mma.1670120603. |
[15] |
K. Dressler, Stationary solutions of the Vlasov-Fokker-Planck equation,, Math. Meth. Appl. Sci., 9 (1987), 169.
doi: 10.1002/mma.1670090113. |
[16] |
J. Dunkel and P. Hänggi, Theory of the relativistic Brownian motion: The (1+3)-dimensional case,, Phys. Rev. E, 72 (2005).
doi: 10.1103/PhysRevE.72.036106. |
[17] |
J. Dunkel and P. Hänggi, Relativistic Brownian motion,, Phys. Rep., 471 (2009), 1.
doi: 10.1016/j.physrep.2008.12.001. |
[18] |
D. T. Frank, "Nonlinear Fokker-Planck Equations: Fundamentals and Applications,", Springer Series in Synergetics {\bf 25}, 25 (2005). Google Scholar |
[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.
doi: 10.1006/jmaa.1996.0360. |
[20] |
Z. Haba, Relativistic diffusion,, Phys. Rev. E, 79 (2009).
doi: 10.1103/PhysRevE.79.021128. |
[21] |
Z. Haba, Relativistic diffusion of elementary particles with spin,, Journ. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/44/445401. |
[22] |
Z. Haba, Energy and entropy of relativistic diffusing particles,, Mod. Phys. Lett. A, 25 (2010), 2683.
doi: 10.1142/S0217732310033992. |
[23] |
Z. Haba, Relativistic diffusive transport,, preprint \arXiv{0911.3126}., (). Google Scholar |
[24] |
B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators And Witten Laplacians,", Lecture Notes in Mathematics {\bf 1862}, 1862 (1862).
|
[25] |
L. Hörmander, Pseudodifferential operators and non-elliptic boundary problems,, Ann. of Math., 83 (1966), 129.
doi: 10.2307/1970473. |
[26] |
F. John, Blow-up for quasi linear wave equations in three space dimensions,, Comm. Pure Appl. Math., 34 (1981), 29.
doi: 10.1002/cpa.3160340103. |
[27] |
I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", Graduate Texts in Mathematics {\bf 113} 2$^{nd}$ edition, 113 (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.
doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G. |
[29] |
E. H. Lieb and M. Loss, "Analysis,", American Math. Soc. {\bf 14}, 14 (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. Google Scholar |
[31] |
M. Risken, "The Fokker-Planck Equation: Methods of Solutions and Applications,", Springer Series in Synergetics {\bf 18}, 18 (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.
doi: 10.1111/j.1365-2966.2006.11215.x. |
[33] |
S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity,, Phys. Rev. D, 47 (1993), 1529.
doi: 10.1103/PhysRevD.47.1529. |
[34] |
C. Sogge, "Lectures on Nonlinear Wave Equations,", International Press, (1995).
|
[35] |
J. L. Vázquez, "The Porous Medium Equation: Mathematical Theory,", Oxford Math. Monogr., (2007). Google Scholar |
[36] |
T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system,, SIAM J. Math. Anal., 42 (2010), 459.
doi: 10.1137/090755796. |
[37] |
C. Villani, Hypocoercivity,, Memoirs of the AMS, 202 (2009). Google Scholar |
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