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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.  Google Scholar

[15]

L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[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.  Google Scholar

[23]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.  Google Scholar

[15]

L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[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.  Google Scholar

[23]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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