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May  2013, 18(3): 681-691. doi: 10.3934/dcdsb.2013.18.681

On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian

Hyung Ju Hwang 1, and  Juhi Jang 2,

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

Received  September 2012 Revised  September 2012 Published  December 2012

We establish the exponential time decay rate of smooth solutions of small amplitude to the Vlasov-Poisson-Fokker-Planck equations to the Maxwellian both in the whole space and in the periodic box via the uniform-in-time energy estimates and also the macroscopic equations.
Keywords: hypocoercivity, exponential time decay rate, Vlasov-Poisson-Fokker-Planck, energy estimates, macroscopic equations..
Mathematics Subject Classification: Primary: 35Q84; Secondary: 82D1.
Citation: Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681
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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