Explicit structure of the Fokker-Planck equation with potential
Authors:
Yu-Chu Lin, Haitao Wang and Kung-Chien Wu
Journal:
Quart. Appl. Math. 77 (2019), 727-766
MSC (2010):
Primary 35Q84, 82C40
DOI:
https://doi.org/10.1090/qam/1537
Published electronically:
March 11, 2019
MathSciNet review:
4009330
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Additional Information
Abstract: We study the pointwise (in the space and time variables) behavior of the Fokker-Planck equation with potential. An explicit description of the solution is given, including the large time behavior, initial layer, and spatially asymptotic behavior. Moreover, it is shown that the structure of the solution sensitively depends on the potential function.
References
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Ana Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci. 21 (1998), no. 11, 985–1014. MR 1635981, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819980725%2921%3A11%3C985%3A%3AAID-MMA919%3E3.0.CO%3B2-B
- Chiun-Chuan Chen, Tai-Ping Liu, and Tong Yang, Existence of boundary layer solutions to the Boltzmann equation, Anal. Appl. (Singap.) 2 (2004), no. 4, 337–363. MR 2103866, DOI https://doi.org/10.1142/S0219530504000400
- Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys. 74 (1980), no. 1, 71–95. MR 575897
- Kleber Carrapatoso, Isabelle Tristani, and Kung-Chien Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 363–418. MR 3483898, DOI https://doi.org/10.1007/s00205-015-0963-x
- P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Rational Mech. Anal. 138 (1997), no. 2, 137–167. MR 1463805, DOI https://doi.org/10.1007/s002050050038
- L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), no. 1, 1–42. MR 1787105, DOI https://doi.org/10.1002/1097-0312%28200101%2954%3A1%3C1%3A%3AAID-CPA1%3E3.0.CO%3B2-Q
- Renjun Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity 24 (2011), no. 8, 2165–2189. MR 2813582, DOI https://doi.org/10.1088/0951-7715/24/8/003
- Fei Fei, Zhaohui Liu, Jun Zhang, and Chuguang Zheng, A particle Fokker-Planck algorithm with multiscale temporal discretization for rarefied and continuum gas flows, Commun. Comput. Phys. 22 (2017), no. 2, 338–374. MR 3668415, DOI https://doi.org/10.4208/cicp.OA-2016-0134
- M. H. Gorji, M. Torrilhon, and P. Jenny, Fokker-Planck model for computational studies of monatomic rarefied gas flows, J. Fluid Mech. 680 (2011), 574–601. MR 2819572, DOI https://doi.org/10.1017/jfm.2011.188
- Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391–434. MR 1946444, DOI https://doi.org/10.1007/s00220-002-0729-9
- Seung-Yeal Ha and Se Eun Noh, Remarks on the stability of the frictionless Vlasov-Poisson-Fokker-Planck system, J. Math. Phys. 48 (2007), no. 7, 073303, 13. MR 2337680, DOI https://doi.org/10.1063/1.2746130
- Frédéric Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal. 244 (2007), no. 1, 95–118 (English, with English and French summaries). MR 2294477, DOI https://doi.org/10.1016/j.jfa.2006.11.013
- Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. MR 2034753, DOI https://doi.org/10.1007/s00205-003-0276-3
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI https://doi.org/10.1007/BF02392081
- Hyung Ju Hwang, Juhi Jang, and Jaewoo Jung, On the kinetic Fokker-Planck equation in a half-space with absorbing barriers, Indiana Univ. Math. J. 64 (2015), no. 6, 1767–1804. MR 3436235, DOI https://doi.org/10.1512/iumj.2015.64.5679
- Hyung Ju Hwang, Juhi Jang, and Jaewoo Jung, The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal. 50 (2018), no. 2, 2194–2232. MR 3788197, DOI https://doi.org/10.1137/16M1109928
- Hyung Ju Hwang and Du Phan, On the Fokker-Planck equations with inflow boundary conditions, Quart. Appl. Math. 75 (2017), no. 2, 287–308. MR 3614499, DOI https://doi.org/10.1090/qam/1462
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math. 7 (1990), no. 2, 301–320. MR 1057534, DOI https://doi.org/10.1007/BF03167846
- A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116–117 (German). MR 1503147, DOI https://doi.org/10.2307/1968123
- Ming-Yi Lee, Tai-Ping Liu, and Shih-Hsien Yu, Large-time behavior of solutions for the Boltzmann equation with hard potentials, Comm. Math. Phys. 269 (2007), no. 1, 17–37. MR 2274461, DOI https://doi.org/10.1007/s00220-006-0108-z
- R. L. Liboff, Kinetic theory: Classical, quantum, and relativistic descriptions, Springer Science and Business Media, 2003.
- Yu-Chu Lin, Haitao Wang, and Kung-Chien Wu, Quantitative pointwise estimate of the solution of the linearized Boltzmann equation, J. Stat. Phys. 171 (2018), no. 5, 927–964. MR 3800900, DOI https://doi.org/10.1007/s10955-018-2047-4
- Tai-Ping Liu and Weike Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys. 196 (1998), no. 1, 145–173. MR 1643525, DOI https://doi.org/10.1007/s002200050418
- Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI https://doi.org/10.1002/cpa.20011
- Tai-Ping Liu and Shih-Hsien Yu, Green’s function of Boltzmann equation, 3-D waves, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 1, 1–78. MR 2230121
- Tai-Ping Liu and Shih-Hsien Yu, Solving Boltzmann equation, Part I: Green’s function, Bull. Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 2, 115–243. MR 2850554
- Lan Luo and Hongjun Yu, Spectrum analysis of the linear Fokker-Planck equation, Anal. Appl. (Singap.) 15 (2017), no. 3, 313–331. MR 3636059, DOI https://doi.org/10.1142/S0219530515500219
- S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal. 221 (2016), no. 2, 677–723. MR 3488535, DOI https://doi.org/10.1007/s00205-016-0972-4
- Clément Mouhot and Lukas Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity 19 (2006), no. 4, 969–998. MR 2214953, DOI https://doi.org/10.1088/0951-7715/19/4/011
- Kosuke Ono and Walter A. Strauss, Regular solutions of the Vlasov-Poisson-Fokker-Planck system, Discrete Contin. Dynam. Systems 6 (2000), no. 4, 751–772. MR 1788250, DOI https://doi.org/10.3934/dcds.2000.6.751
- Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287–339. MR 2366140, DOI https://doi.org/10.1007/s00205-007-0067-3
- Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models 5 (2012), no. 3, 583–613. MR 2972454, DOI https://doi.org/10.3934/krm.2012.5.583
- Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
- Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141. MR 2562709, DOI https://doi.org/10.1090/S0065-9266-09-00567-5
- Hassler Whitney, Differentiable even functions, Duke Math. J. 10 (1943), 159–160. MR 7783
- Tong Yang and Hongjun Yu, Spectrum analysis of some kinetic equations, Arch. Ration. Mech. Anal. 222 (2016), no. 2, 731–768. MR 3544316, DOI https://doi.org/10.1007/s00205-016-1010-2
References
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Ana Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci. 21 (1998), no. 11, 985–1014. MR 1635981, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819980725%2921%3A11%24%5Clangle%24985%3A%3AAID-MMA919%24%5Crangle%243.0.CO%3B2-B
- Chiun-Chuan Chen, Tai-Ping Liu, and Tong Yang, Existence of boundary layer solutions to the Boltzmann equation, Anal. Appl. (Singap.) 2 (2004), no. 4, 337–363. MR 2103866, DOI https://doi.org/10.1142/S0219530504000400
- Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys. 74 (1980), no. 1, 71–95. MR 575897
- Kleber Carrapatoso, Isabelle Tristani, and Kung-Chien Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 363–418. MR 3483898, DOI https://doi.org/10.1007/s00205-015-0963-x
- P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Rational Mech. Anal. 138 (1997), no. 2, 137–167. MR 1463805, DOI https://doi.org/10.1007/s002050050038
- L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), no. 1, 1–42. MR 1787105, DOI https://doi.org/10.1002/1097-0312%28200101%2954%3A1%24%5Clangle%241%3A%3AAID-CPA1%24%5Crangle%243.0.CO%3B2-Q
- Renjun Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity 24 (2011), no. 8, 2165–2189. MR 2813582, DOI https://doi.org/10.1088/0951-7715/24/8/003
- Fei Fei, Zhaohui Liu, Jun Zhang, and Chuguang Zheng, A particle Fokker-Planck algorithm with multiscale temporal discretization for rarefied and continuum gas flows, Commun. Comput. Phys. 22 (2017), no. 2, 338–374. MR 3668415, DOI https://doi.org/10.4208/cicp.OA-2016-0134
- M. H. Gorji, M. Torrilhon, and P. Jenny, Fokker-Planck model for computational studies of monatomic rarefied gas flows, J. Fluid Mech. 680 (2011), 574–601. MR 2819572, DOI https://doi.org/10.1017/jfm.2011.188
- Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391–434. MR 1946444, DOI https://doi.org/10.1007/s00220-002-0729-9
- Seung-Yeal Ha and Se Eun Noh, Remarks on the stability of the frictionless Vlasov-Poisson-Fokker-Planck system, J. Math. Phys. 48 (2007), no. 7, 073303, 13. MR 2337680, DOI https://doi.org/10.1063/1.2746130
- Frédéric Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal. 244 (2007), no. 1, 95–118 (English, with English and French summaries). MR 2294477, DOI https://doi.org/10.1016/j.jfa.2006.11.013
- Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. MR 2034753, DOI https://doi.org/10.1007/s00205-003-0276-3
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 0222474, DOI https://doi.org/10.1007/BF02392081
- Hyung Ju Hwang, Juhi Jang, and Jaewoo Jung, On the kinetic Fokker-Planck equation in a half-space with absorbing barriers, Indiana Univ. Math. J. 64 (2015), no. 6, 1767–1804. MR 3436235, DOI https://doi.org/10.1512/iumj.2015.64.5679
- Hyung Ju Hwang, Juhi Jang, and Jaewoo Jung, The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal. 50 (2018), no. 2, 2194–2232. MR 3788197, DOI https://doi.org/10.1137/16M1109928
- Hyung Ju Hwang and Du Phan, On the Fokker-Planck equations with inflow boundary conditions, Quart. Appl. Math. 75 (2017), no. 2, 287–308. MR 3614499, DOI https://doi.org/10.1090/qam/1462
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math. 7 (1990), no. 2, 301–320. MR 1057534, DOI https://doi.org/10.1007/BF03167846
- A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116–117 (German). MR 1503147, DOI https://doi.org/10.2307/1968123
- Ming-Yi Lee, Tai-Ping Liu, and Shih-Hsien Yu, Large-time behavior of solutions for the Boltzmann equation with hard potentials, Comm. Math. Phys. 269 (2007), no. 1, 17–37. MR 2274461, DOI https://doi.org/10.1007/s00220-006-0108-z
- R. L. Liboff, Kinetic theory: Classical, quantum, and relativistic descriptions, Springer Science and Business Media, 2003.
- Yu-Chu Lin, Haitao Wang, and Kung-Chien Wu, Quantitative pointwise estimate of the solution of the linearized Boltzmann equation, J. Stat. Phys. 171 (2018), no. 5, 927–964. MR 3800900, DOI https://doi.org/10.1007/s10955-018-2047-4
- Tai-Ping Liu and Weike Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys. 196 (1998), no. 1, 145–173. MR 1643525, DOI https://doi.org/10.1007/s002200050418
- Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI https://doi.org/10.1002/cpa.20011
- Tai-Ping Liu and Shih-Hsien Yu, Green’s function of Boltzmann equation, 3-D waves, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 1, 1–78. MR 2230121
- Tai-Ping Liu and Shih-Hsien Yu, Solving Boltzmann equation, Part I: Green’s function, Bull. Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 2, 115–243. MR 2850554
- Lan Luo and Hongjun Yu, Spectrum analysis of the linear Fokker-Planck equation, Anal. Appl. (Singap.) 15 (2017), no. 3, 313–331. MR 3636059, DOI https://doi.org/10.1142/S0219530515500219
- S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal. 221 (2016), no. 2, 677–723. MR 3488535, DOI https://doi.org/10.1007/s00205-016-0972-4
- Clément Mouhot and Lukas Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity 19 (2006), no. 4, 969–998. MR 2214953, DOI https://doi.org/10.1088/0951-7715/19/4/011
- Kosuke Ono and Walter A. Strauss, Regular solutions of the Vlasov-Poisson-Fokker-Planck system, Discrete Contin. Dynam. Systems 6 (2000), no. 4, 751–772. MR 1788250, DOI https://doi.org/10.3934/dcds.2000.6.751
- Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287–339. MR 2366140, DOI https://doi.org/10.1007/s00205-007-0067-3
- Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models 5 (2012), no. 3, 583–613. MR 2972454, DOI https://doi.org/10.3934/krm.2012.5.583
- Michael E. Taylor, Partial differential equations. III, Nonlinear equations, corrected reprint of the 1996 original, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. MR 1477408
- Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141. MR 2562709, DOI https://doi.org/10.1090/S0065-9266-09-00567-5
- Hassler Whitney, Differentiable even functions, Duke Math. J. 10 (1943), 159–160. MR 0007783
- Tong Yang and Hongjun Yu, Spectrum analysis of some kinetic equations, Arch. Ration. Mech. Anal. 222 (2016), no. 2, 731–768. MR 3544316, DOI https://doi.org/10.1007/s00205-016-1010-2
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Additional Information
Yu-Chu Lin
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan, Taiwan
MR Author ID:
843221
Email:
yuchu@mail.ncku.edu.tw
Haitao Wang
Affiliation:
Institute of Natural Sciences and School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, People’s Republic of China
MR Author ID:
1050454
Email:
haitallica@sjtu.edu.cn
Kung-Chien Wu
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan, Taiwan; and National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan
MR Author ID:
887455
Email:
kungchienwu@gmail.com
Keywords:
Fokker-Planck,
fluid-like waves,
kinetic-like waves,
pointwise estimate,
regularization estimate
Received by editor(s):
October 27, 2018
Received by editor(s) in revised form:
January 24, 2019
Published electronically:
March 11, 2019
Additional Notes:
The first author was supported by the Ministry of Science and Technology under the grant MOST 107-2115-M-006-006-.
The second author was sponsored by Shanghai Sailing Program(18YF1411800) and Shanghai Jiao Tong University(WF220441907).
The third author was supported by the Ministry of Science and Technology under the grant 108-2636-M-006-005- and National Center for Theoretical Sciences.
Article copyright:
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