June  2022, 15(3): 355-384. doi: 10.3934/krm.2021052

Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system

1. 

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

2. 

Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea

3. 

Department of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

*Corresponding author: José A. Carrillo

Received  March 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

We provide a quantitative asymptotic analysis for the nonlinear Vlasov–Poisson–Fokker–Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov–Poisson–Fokker–Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.

Citation: José A. Carrillo, Young-Pil Choi, Yingping Peng. Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system. Kinetic and Related Models, 2022, 15 (3) : 355-384. doi: 10.3934/krm.2021052
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[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.

[3]

J. A. Carrillo and Y.-P. Choi, Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 925-954.  doi: 10.1016/j.anihpc.2020.02.001.

[4]

J. A. CarrilloY.-P. Choi and J. Jung, Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces, Math. Models Methods Appl. Sci., 31 (2021), 327-408.  doi: 10.1142/S0218202521500081.

[5]

J. A. CarrilloY.-P. Choi and O. Tse, Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces, Commun. Math. Phys., 365 (2019), 329-361.  doi: 10.1007/s00220-018-3276-8.

[6]

J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, In Active Particles, Modelling and Simulation in Science and Technology, 2 (2019), 65–108.

[7]

J. A. CarrilloE. FeireislP. Gwiazda and A. öwierczewska-Gwiazda, Weak solutions for Euler systems with non-local interactions, J. Lond. Math. Soc., 95 (2017), 705-724.  doi: 10.1112/jlms.12027.

[8]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376.

[9]

J. A. CarrilloY. Peng and A. Wróblewska-Kamińska, Relative entropy method for the relaxation limit of hydrodynamic models, Netw. Heterog. Media, 15 (2020), 369-387.  doi: 10.3934/nhm.2020023.

[10]

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.

[11]

J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker–Planck-type equations, Math. Methods Appl. Sci., 21 (1998), 1269-1286.  doi: 10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.0.CO;2-O.

[12]

Y.-P. Choi, Global classical solutions of the Vlasov–Fokker–Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887.

[13]

Y.-P. Choi, Large friction limit of pressureless Euler equations with nonlocal forces, J. Differential Equations, 299 (2021), 196-228.  doi: 10.1016/j.jde.2021.07.024.

[14]

Y.-P. Choi and I.-J. Jeong, Classical solutions for fractional porous medium flow, Nonlinear Anal., 210 (2021), Paper No. 112393, 13 pp. doi: 10.1016/j.na.2021.112393.

[15]

Y.-P. Choi and I.-J. Jeong, Relaxation to the fractional porous medium equation from the Euler–Riesz system, J. Nonlinear Sci., 31 (2021), Article no. 95, 28 pp. doi: 10.1007/s00332-021-09754-w.

[16]

Y.-P. Choi and O. Tse, Quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with singular interaction forces, preprint, arXiv: 2012.00422.

[17]

J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Trans. Amer. Math. Soc., 359 (2007), 637-648.  doi: 10.1090/S0002-9947-06-04028-1.

[18]

P. Degond, Global existence of smooth solutions for the Vlasov–Fokker–Planck equation in 1 and 2 space dimensions, Ann Sci. École Norm. Sup., 19 (1986), 519–542. doi: 10.24033/asens.1516.

[19]

M. H. DuongA. LamaczM. A. PeletierA. Schlichting and U. Sharma, Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics, Nonlinearity, 31 (2018), 4517-4566.  doi: 10.1088/1361-6544/aaced5.

[20]

M. H. Duong, A. Lamacz, M. A. Peletier and U. Sharma, Variational approach to coarse-graining of generalized gradient flows, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 100, 65 pp. doi: 10.1007/s00526-017-1186-9.

[21]

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.

[22]

R. Fetecau and W. Sun, First-order aggregation models and zero inertia limits, J. Differential Equations, 259 (2015), 6774-6802.  doi: 10.1016/j.jde.2015.08.018.

[23]

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.

[24]

T. GoudonJ. NietoF. 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.

[25]

P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672.  doi: 10.1016/S0294-1449(00)00118-9.

[26]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[27]

T. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.

[28]

T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker–Smale model, Hyperbolic conservation Laws and Related Analysis with Applications, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227–242. doi: 10.1007/978-3-642-39007-4_11.

[29]

T. K. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker–Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[30]

C. Lattanzio and A. E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal., 45 (2013), 1563-1584.  doi: 10.1137/120891307.

[31]

C. Lattanzio and A. E. Tzavaras, From gas dynamics with large friction to gradient flows describing diffusion theories, Comm. Partial Differential Equations, 42 (2017), 261-290.  doi: 10.1080/03605302.2016.1269808.

[32]

J. NietoF. 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.

[33]

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

[34]

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.

[35]

H. D. VictoryJr . 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.

[36]

C. Villani, A review of mathematical topics in collisional kinetic theory, In Handbook of Mathematical Fluid Dynamics, Amsterdam: North-Holland, 1 (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[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.

[3]

J. A. Carrillo and Y.-P. Choi, Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 925-954.  doi: 10.1016/j.anihpc.2020.02.001.

[4]

J. A. CarrilloY.-P. Choi and J. Jung, Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces, Math. Models Methods Appl. Sci., 31 (2021), 327-408.  doi: 10.1142/S0218202521500081.

[5]

J. A. CarrilloY.-P. Choi and O. Tse, Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces, Commun. Math. Phys., 365 (2019), 329-361.  doi: 10.1007/s00220-018-3276-8.

[6]

J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, In Active Particles, Modelling and Simulation in Science and Technology, 2 (2019), 65–108.

[7]

J. A. CarrilloE. FeireislP. Gwiazda and A. öwierczewska-Gwiazda, Weak solutions for Euler systems with non-local interactions, J. Lond. Math. Soc., 95 (2017), 705-724.  doi: 10.1112/jlms.12027.

[8]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376.

[9]

J. A. CarrilloY. Peng and A. Wróblewska-Kamińska, Relative entropy method for the relaxation limit of hydrodynamic models, Netw. Heterog. Media, 15 (2020), 369-387.  doi: 10.3934/nhm.2020023.

[10]

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.

[11]

J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker–Planck-type equations, Math. Methods Appl. Sci., 21 (1998), 1269-1286.  doi: 10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.0.CO;2-O.

[12]

Y.-P. Choi, Global classical solutions of the Vlasov–Fokker–Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887.

[13]

Y.-P. Choi, Large friction limit of pressureless Euler equations with nonlocal forces, J. Differential Equations, 299 (2021), 196-228.  doi: 10.1016/j.jde.2021.07.024.

[14]

Y.-P. Choi and I.-J. Jeong, Classical solutions for fractional porous medium flow, Nonlinear Anal., 210 (2021), Paper No. 112393, 13 pp. doi: 10.1016/j.na.2021.112393.

[15]

Y.-P. Choi and I.-J. Jeong, Relaxation to the fractional porous medium equation from the Euler–Riesz system, J. Nonlinear Sci., 31 (2021), Article no. 95, 28 pp. doi: 10.1007/s00332-021-09754-w.

[16]

Y.-P. Choi and O. Tse, Quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with singular interaction forces, preprint, arXiv: 2012.00422.

[17]

J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Trans. Amer. Math. Soc., 359 (2007), 637-648.  doi: 10.1090/S0002-9947-06-04028-1.

[18]

P. Degond, Global existence of smooth solutions for the Vlasov–Fokker–Planck equation in 1 and 2 space dimensions, Ann Sci. École Norm. Sup., 19 (1986), 519–542. doi: 10.24033/asens.1516.

[19]

M. H. DuongA. LamaczM. A. PeletierA. Schlichting and U. Sharma, Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics, Nonlinearity, 31 (2018), 4517-4566.  doi: 10.1088/1361-6544/aaced5.

[20]

M. H. Duong, A. Lamacz, M. A. Peletier and U. Sharma, Variational approach to coarse-graining of generalized gradient flows, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 100, 65 pp. doi: 10.1007/s00526-017-1186-9.

[21]

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.

[22]

R. Fetecau and W. Sun, First-order aggregation models and zero inertia limits, J. Differential Equations, 259 (2015), 6774-6802.  doi: 10.1016/j.jde.2015.08.018.

[23]

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.

[24]

T. GoudonJ. NietoF. 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.

[25]

P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672.  doi: 10.1016/S0294-1449(00)00118-9.

[26]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[27]

T. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.

[28]

T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker–Smale model, Hyperbolic conservation Laws and Related Analysis with Applications, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227–242. doi: 10.1007/978-3-642-39007-4_11.

[29]

T. K. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker–Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[30]

C. Lattanzio and A. E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal., 45 (2013), 1563-1584.  doi: 10.1137/120891307.

[31]

C. Lattanzio and A. E. Tzavaras, From gas dynamics with large friction to gradient flows describing diffusion theories, Comm. Partial Differential Equations, 42 (2017), 261-290.  doi: 10.1080/03605302.2016.1269808.

[32]

J. NietoF. 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.

[33]

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

[34]

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.

[35]

H. D. VictoryJr . 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.

[36]

C. Villani, A review of mathematical topics in collisional kinetic theory, In Handbook of Mathematical Fluid Dynamics, Amsterdam: North-Holland, 1 (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0.

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