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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
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show all references

References:
[1]

J. Math. Pures Appl. (9), 81 (2002), 1135-1159. doi: 10.1016/S0021-7824(02)01264-3.  Google Scholar

[2]

J. Funct. Anal., 111 (1993), 239-258. doi: 10.1006/jfan.1993.1011.  Google Scholar

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J. Differential Equations, 122 (1995), 225-238. doi: 10.1006/jdeq.1995.1146.  Google Scholar

[4]

Differential Integral Equations, 8 (1995), 487-514.  Google Scholar

[5]

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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Kinet. Relat. Models, 4 (2011), 227-258. doi: 10.3934/krm.2011.4.227.  Google Scholar

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Math. Methods Appl. Sci., 18 (1995), 825-839. doi: 10.1002/mma.1670181006.  Google Scholar

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J. Funct. Anal., 141 (1996), 99-132. doi: 10.1006/jfan.1996.0123.  Google Scholar

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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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Arch. Ration. Mech. Anal., 195 (2010), 75-116. doi: 10.1007/s00205-008-0184-7.  Google Scholar

[13]

SIAM J. Math. Anal., 42 (2010), 2177-2202. doi: 10.1137/090776755.  Google Scholar

[14]

Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.  Google Scholar

[15]

Acta Math., 119 (1967), 147-171.  Google Scholar

[16]

Arch. Ration. Mech. Anal., 171 (2004), 151-218. doi: 10.1007/s00205-003-0276-3.  Google Scholar

[17]

Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar

[18]

Comm. Math. Phys., 261 (2006), 629-672. doi: 10.1007/s00220-005-1455-x.  Google Scholar

[19]

Discrete Contin. Dynam. Systems, 6 (2000), 751-772. doi: 10.3934/dcds.2000.6.751.  Google Scholar

[20]

J. Differential Equations, 99 (1992), 59-77. doi: 10.1016/0022-0396(92)90135-A.  Google Scholar

[21]

Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

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[23]

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

[24]

J. Math. Anal. Appl., 160 (1991), 525-555. doi: 10.1016/0022-247X(91)90324-S.  Google Scholar

[25]

Indiana Univ. Math. J., 39 (1990), 105-156. doi: 10.1512/iumj.1990.39.39009.  Google Scholar

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