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doi: 10.3934/krm.2021041
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Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system

Mingying Zhong ,

College of Mathematics and Information Sciences, Guangxi University, China

*Corresponding author: Mingying Zhong

Received  March 2021 Revised  August 2021 Early access December 2021

Fund Project: The research of the author was supported by the National Science Fund for Excellent Young Scholars No. 11922107, the National Natural Science Foundation of China grants No. 11671100, and Guangxi Natural Science Foundation Nos. 2018GXNSFAA138210 and 2019JJG110010

In the present paper, we study the diffusion limit of the classical solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system with initial data near a global Maxwellian. We prove the convergence and establish the optimal convergence rate of the global strong solution to the VPFP system towards the solution to the drift-diffusion-Poisson system based on the spectral analysis with precise estimation on the initial layer.

Keywords: Vlasov-Poisson-Fokker-Planck system, spectral analysis, diffusion limit, convergence rate.
Mathematics Subject Classification: 76P05, 82C40, 82D05.
Citation: Mingying Zhong. Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, doi: 10.3934/krm.2021041
References:
[1]

A. Arnold, P. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems, Transp. Theory Statist. Phys., 29 (2000), 571-581.  doi: 10.1080/00411450008205893.  Google Scholar

[2]

C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257.  doi: 10.1142/S0218202591000137.  Google Scholar

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P. Biler and J. Dolbeault, Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré, 1 (2000), 461-472.  doi: 10.1007/S000230050003.  Google Scholar

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P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

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L. L. Bonilla, J. A. Carrillo and J. Soler, Asymptotic behavior of an initial-boundary value problem for the Vlasov-Poisson-Fokker-Planck system, SIAM J. Appl. Math., 57 (1997), 1343-1372.  doi: 10.1137/S0036139995291544.  Google Scholar

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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

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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

[9]

F. Bouchut and J. Dolbeault, On long 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

[10]

J. A. Carrillo, Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system, Math. Methods Appl. Sci., 21 (1998), 907-938.  doi: 10.1002/(SICI)1099-1476(19980710)21:10<907::AID-MMA977>3.0.CO;2-W.  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

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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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N. El Ghani and N. Masmoudi, Diffusion limit of the Vlasov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8 (2010), 463-479.  doi: 10.4310/CMS.2010.v8.n2.a9.  Google Scholar

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W. Fang and K. Ito, On the time-dependent drift-diffusion model for semiconductors, J. Differential Equations, 117 (1995), 245-280.  doi: 10.1006/jdeq.1995.1054.  Google Scholar

[16]

T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15 (2005), 737-752.  doi: 10.1142/S021820250500056X.  Google Scholar

[17]

T. Goudon, J. Nieto, F. Poupaud and J. Soler, Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, J. Differential Equations, 213 (2005), 418-442.  doi: 10.1016/j.jde.2004.09.008.  Google Scholar

[18]

H. J. Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681-691.  doi: 10.3934/dcdsb.2013.18.681.  Google Scholar

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A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations and their Applications, 41, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8334-4.  Google Scholar

[20]

A. Krzywicki and T. A. Nadzieja, A nonstationary problem in the theory of electrolytes, Quart. Appl. Math., 50 (1992), 105-107.  doi: 10.1090/qam/1146626.  Google Scholar

[21]

H.-L. Li, T. Yang, J. Sun and M. Zhong, Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker-Planck) equations, Sci. Sin. Math., 46 (2016), 981-1004.  doi: 10.1360/N012015-00230.  Google Scholar

[22]

H.-L. Li, T. Yang and M. Zhong, Diffusion limit of the Vlasov-Poisson-Boltzmann system, Kinet. Relat. Models, 14 (2021), 211-255.  doi: 10.3934/krm.2021003.  Google Scholar

[23]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, Arch. Ration. Mech. Anal., 241 (2021), 311-355.  doi: 10.1007/s00205-021-01652-5.  Google Scholar

[24]

L. Luo and H. Yu, Spectrum analysis of the linear Fokker-Planck equation, Anal. Appl. (Singap.), 15 (2017), 313-331.  doi: 10.1142/S0219530515500219.  Google Scholar

[25]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[26]

J. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Phil. Trans. Roy. Soc. Lond., 170 (1879), 231-256.   Google Scholar

[27]

J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 158 (2001), 29-59.  doi: 10.1007/s002050100139.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10 (2000), 1027-1045.  doi: 10.1142/S0218202500000525.  Google Scholar

[30]

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

[31]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Series - No. 8, Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, 2006. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.110&rep=rep1&type=pdf. Google Scholar

[32]

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

[33]

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

[34]

H. Wu, T.-C. Lin and C. Liu, Diffusion limit of kinetic equations for multiple species charged particles, Arch. Ration. Mech. Anal., 215 (2015), 419-441.  doi: 10.1007/s00205-014-0784-3.  Google Scholar

show all references

References:
[1]

A. Arnold, P. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems, Transp. Theory Statist. Phys., 29 (2000), 571-581.  doi: 10.1080/00411450008205893.  Google Scholar

[2]

C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257.  doi: 10.1142/S0218202591000137.  Google Scholar

[3]

P. Biler, Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions, Nonlinear Anal., 19 (1992), 1121-1136.  doi: 10.1016/0362-546X(92)90186-I.  Google Scholar

[4]

P. Biler and J. Dolbeault, Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré, 1 (2000), 461-472.  doi: 10.1007/S000230050003.  Google Scholar

[5]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[6]

L. L. Bonilla, J. A. Carrillo and J. Soler, Asymptotic behavior of an initial-boundary value problem for the Vlasov-Poisson-Fokker-Planck system, SIAM J. Appl. Math., 57 (1997), 1343-1372.  doi: 10.1137/S0036139995291544.  Google Scholar

[7]

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

[8]

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

[9]

F. Bouchut and J. Dolbeault, On long 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

[10]

J. A. Carrillo, Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system, Math. Methods Appl. Sci., 21 (1998), 907-938.  doi: 10.1002/(SICI)1099-1476(19980710)21:10<907::AID-MMA977>3.0.CO;2-W.  Google Scholar

[11]

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

[12]

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

[13]

N. El Ghani and N. Masmoudi, Diffusion limit of the Vlasov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8 (2010), 463-479.  doi: 10.4310/CMS.2010.v8.n2.a9.  Google Scholar

[14]

R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9), 54 (1975), 125-156.   Google Scholar

[15]

W. Fang and K. Ito, On the time-dependent drift-diffusion model for semiconductors, J. Differential Equations, 117 (1995), 245-280.  doi: 10.1006/jdeq.1995.1054.  Google Scholar

[16]

T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15 (2005), 737-752.  doi: 10.1142/S021820250500056X.  Google Scholar

[17]

T. Goudon, J. Nieto, F. Poupaud and J. Soler, Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, J. Differential Equations, 213 (2005), 418-442.  doi: 10.1016/j.jde.2004.09.008.  Google Scholar

[18]

H. J. Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681-691.  doi: 10.3934/dcdsb.2013.18.681.  Google Scholar

[19]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations and their Applications, 41, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8334-4.  Google Scholar

[20]

A. Krzywicki and T. A. Nadzieja, A nonstationary problem in the theory of electrolytes, Quart. Appl. Math., 50 (1992), 105-107.  doi: 10.1090/qam/1146626.  Google Scholar

[21]

H.-L. Li, T. Yang, J. Sun and M. Zhong, Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker-Planck) equations, Sci. Sin. Math., 46 (2016), 981-1004.  doi: 10.1360/N012015-00230.  Google Scholar

[22]

H.-L. Li, T. Yang and M. Zhong, Diffusion limit of the Vlasov-Poisson-Boltzmann system, Kinet. Relat. Models, 14 (2021), 211-255.  doi: 10.3934/krm.2021003.  Google Scholar

[23]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, Arch. Ration. Mech. Anal., 241 (2021), 311-355.  doi: 10.1007/s00205-021-01652-5.  Google Scholar

[24]

L. Luo and H. Yu, Spectrum analysis of the linear Fokker-Planck equation, Anal. Appl. (Singap.), 15 (2017), 313-331.  doi: 10.1142/S0219530515500219.  Google Scholar

[25]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[26]

J. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Phil. Trans. Roy. Soc. Lond., 170 (1879), 231-256.   Google Scholar

[27]

J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 158 (2001), 29-59.  doi: 10.1007/s002050100139.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10 (2000), 1027-1045.  doi: 10.1142/S0218202500000525.  Google Scholar

[30]

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

[31]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Series - No. 8, Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, 2006. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.110&rep=rep1&type=pdf. Google Scholar

[32]

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

[33]

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

[34]

H. Wu, T.-C. Lin and C. Liu, Diffusion limit of kinetic equations for multiple species charged particles, Arch. Ration. Mech. Anal., 215 (2015), 419-441.  doi: 10.1007/s00205-014-0784-3.  Google Scholar

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