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February  2022, 15(1): 1-26. doi: 10.3934/krm.2021041

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 Published  February 2022 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 and Related Models, 2022, 15 (1) : 1-26. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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