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On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian
1. | Department of Mathematics, Pohang University of Science and Technology, Pohang, Gyeongbuk |
2. | Department of Mathematics, University of California, Riverside, Riverside, CA 92521, United States |
References:
[1] |
F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pures Appl. (9), 81 (2002), 1135-1159.
doi: 10.1016/S0021-7824(02)01264-3. |
[2] |
F. Bouchut, Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal., 111 (1993), 239-258.
doi: 10.1006/jfan.1993.1011. |
[3] |
F. Bouchut, Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations, 122 (1995), 225-238.
doi: 10.1006/jdeq.1995.1146. |
[4] |
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, Differential Integral Equations, 8 (1995), 487-514. |
[5] |
A. Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci., 21 (1998), 985-1014.
doi: 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B. |
[6] |
J. Carrillo, R. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system, Kinet. Relat. Models, 4 (2011), 227-258.
doi: 10.3934/krm.2011.4.227. |
[7] |
J. A. Carrillo and J. Soler, On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in $L^p$ spaces, Math. Methods Appl. Sci., 18 (1995), 825-839.
doi: 10.1002/mma.1670181006. |
[8] |
J. A. Carrillo, J. Soler and J. L. Vazquez, Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system, J. Funct. Anal., 141 (1996), 99-132.
doi: 10.1006/jfan.1996.0123. |
[9] |
L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.
doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. |
[10] |
L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[11] |
R. Duan and R. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[12] |
R. Esposito, Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics, Arch. Ration. Mech. Anal., 195 (2010), 75-116.
doi: 10.1007/s00205-008-0184-7. |
[13] |
T. Goudon, L. He, A. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202.
doi: 10.1137/090776755. |
[14] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[15] |
L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. |
[16] |
F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
doi: 10.1007/s00205-003-0276-3. |
[17] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[18] |
C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672.
doi: 10.1007/s00220-005-1455-x. |
[19] |
K. Ono and W. A. Strauss, Regular solutions of the Vlasov-Poisson-Fokker-Planck system, Discrete Contin. Dynam. Systems, 6 (2000), 751-772.
doi: 10.3934/dcds.2000.6.751. |
[20] |
G. Rein and J. Weckler, Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions, J. Differential Equations, 99 (1992), 59-77.
doi: 10.1016/0022-0396(92)90135-A. |
[21] |
E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[22] |
R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[23] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[24] |
H. D. Victory, Jr., On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems, J. Math. Anal. Appl., 160 (1991), 525-555.
doi: 10.1016/0022-247X(91)90324-S. |
[25] |
H. D. Victory, Jr. and B. P. O'Dwyer, On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana Univ. Math. J., 39 (1990), 105-156.
doi: 10.1512/iumj.1990.39.39009. |
show all references
References:
[1] |
F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pures Appl. (9), 81 (2002), 1135-1159.
doi: 10.1016/S0021-7824(02)01264-3. |
[2] |
F. Bouchut, Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal., 111 (1993), 239-258.
doi: 10.1006/jfan.1993.1011. |
[3] |
F. Bouchut, Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations, 122 (1995), 225-238.
doi: 10.1006/jdeq.1995.1146. |
[4] |
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, Differential Integral Equations, 8 (1995), 487-514. |
[5] |
A. Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci., 21 (1998), 985-1014.
doi: 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B. |
[6] |
J. Carrillo, R. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system, Kinet. Relat. Models, 4 (2011), 227-258.
doi: 10.3934/krm.2011.4.227. |
[7] |
J. A. Carrillo and J. Soler, On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in $L^p$ spaces, Math. Methods Appl. Sci., 18 (1995), 825-839.
doi: 10.1002/mma.1670181006. |
[8] |
J. A. Carrillo, J. Soler and J. L. Vazquez, Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system, J. Funct. Anal., 141 (1996), 99-132.
doi: 10.1006/jfan.1996.0123. |
[9] |
L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.
doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. |
[10] |
L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[11] |
R. Duan and R. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[12] |
R. Esposito, Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics, Arch. Ration. Mech. Anal., 195 (2010), 75-116.
doi: 10.1007/s00205-008-0184-7. |
[13] |
T. Goudon, L. He, A. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202.
doi: 10.1137/090776755. |
[14] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[15] |
L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. |
[16] |
F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
doi: 10.1007/s00205-003-0276-3. |
[17] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[18] |
C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672.
doi: 10.1007/s00220-005-1455-x. |
[19] |
K. Ono and W. A. Strauss, Regular solutions of the Vlasov-Poisson-Fokker-Planck system, Discrete Contin. Dynam. Systems, 6 (2000), 751-772.
doi: 10.3934/dcds.2000.6.751. |
[20] |
G. Rein and J. Weckler, Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions, J. Differential Equations, 99 (1992), 59-77.
doi: 10.1016/0022-0396(92)90135-A. |
[21] |
E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[22] |
R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[23] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[24] |
H. D. Victory, Jr., On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems, J. Math. Anal. Appl., 160 (1991), 525-555.
doi: 10.1016/0022-247X(91)90324-S. |
[25] |
H. D. Victory, Jr. and B. P. O'Dwyer, On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana Univ. Math. J., 39 (1990), 105-156.
doi: 10.1512/iumj.1990.39.39009. |
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