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Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system
College of Mathematics and Information Sciences, Guangxi University, China |
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.
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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