doi: 10.3934/krm.2022021
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Global-in-time existence of weak solutions for Vlasov-Manev-Fokker-Planck system

1. 

Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea

2. 

Department of Mathematics and RIM, Seoul National University, Seoul 08826, Republic of Korea

*Corresponding author: In-Jee Jeong

Received  February 2022 Revised  May 2022 Early access July 2022

Fund Project: YPC has been supported by NRF grant (No. 2017R1C1B2012918 and 2022R1A2C100282011) and Yonsei University Research Fund of 2021-22-0301. IJJ has been supported by the New Faculty Startup Fund from Seoul National University and the National Research Foundation of Korea grant (No. 2019R1F1A1058486 and No. 2022R1C1C1011051)

We consider the Vlasov–Manev–Fokker–Planck (VMFP) system in three dimensions, which differs from the Vlasov–Poisson–Fokker–Planck in that it has the gravitational potential of the form $ -1/r - 1/r^2 $ instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions under smallness assumption on either the initial mass or the coefficient of the pure Manev potential. The proof extends to several related kinetic systems.

Citation: Young-Pil Choi, In-Jee Jeong. Global-in-time existence of weak solutions for Vlasov-Manev-Fokker-Planck system. Kinetic and Related Models, doi: 10.3934/krm.2022021
References:
[1]

A. A. Arsen'ev, Global existence of a weak solution of Vlasov's system of equations, USSR Comput. Math. Math. Phys., 15 (1975), 131-143.  doi: 10.1016/0041-5553(75)90141-X.

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/s0294-1449(16)30405-x.

[3]

A. V. BobylevP. DukesR. Illner and H. D. Victory Jr., On Vlasov-Manev equations. Ⅰ. Foundations, properties, and nonglobal existence, J. Statist. Phys., 88 (1997), 885-911.  doi: 10.1023/B:JOSS.0000015177.60491.3c.

[4]

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.

[5]

F. Bouchut and J. Dolbeault, On long time 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. 

[6]

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.

[7]

J. A. Carrillo, Y.-P. Choi and S. Salem, Propagation of chaos for the Vlasov-Poisson-Fokker-Planck equation with a polynomial cut-off, Commun. Contemp. Math., 21 (2019), 28 pp. doi: 10.1142/S0219199718500396.

[8]

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.

[9]

J. A. CarrilloJ. Soler and J. L. Vázquez, Asymptotic behaviour for the frictionless Vlasov-Poisson-Fokker-Planck system, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1195-1200. 

[10]

Y.-P. Choi and I.-J. Jeong, Well-posedness and singularity formation for Vlasov-Riesz system, preprint, 2022, arXiv: 2201.12988.

[11]

P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions, Ann. Sci. École Norm. Sup. (4), 19 (1986), 519–542. doi: 10.24033/asens.1516.

[12]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[13]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791-817.  doi: 10.1016/S0021-7824(99)00021-5.

[14]

E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. Ⅰ. General theory, Math. Methods Appl. Sci., 3 (1981), 229-248.  doi: 10.1002/mma.1670030117.

[15]

E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. Ⅱ. Special cases, Math. Methods Appl. Sci., 4 (1982), 19-32.  doi: 10.1002/mma.1670040104.

[16]

E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279.  doi: 10.1002/mma.1670060118.

[17]

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.

[18]

R. IllnerH. D. VictoryP. Dukes and A. V. Bobylev, On Vlasov-Manev equations. Ⅱ. Local existence and uniqueness, J. Statist. Phys., 91 (1998), 625-654.  doi: 10.1023/A:1023029711405.

[19]

T. K. 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.

[20]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.

[21]

G. Maneff, Die gravitation und das prinzip von wirkung und gegenwirkung, Z. Physik, 31 (1925), 786-802.  doi: 10.1007/BF02980633.

[22]

G. Maneff, La gravitation et l'énergie au zéro, Comptes Rendues, 190 (1930), 1374-1377. 

[23]

G. Maneff, La gravitation et le principle de l'égalité de l'action et de la réaction, Comptes Rendues, 178 (1924), 2159-2161. 

[24]

G. Maneff, Le principe de la moindre action et la gravitation, Comptes Rendues, 190 (1930), 963-965. 

[25]

C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Comm. Partial Differential Equations, 37 (2012), 1273-1285.  doi: 10.1080/03605302.2011.606863.

[26]

B. Perthame and P. E. Souganidis, A limiting case for velocity averaging, Ann. Sci. École Norm. Sup. (4), 31 (1998), 591–598. doi: 10.1016/S0012-9593(98)80108-0.

[27]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[28]

S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka Math. J., 15 (1978), 245-261. 

[29]

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.

[30]

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.

show all references

References:
[1]

A. A. Arsen'ev, Global existence of a weak solution of Vlasov's system of equations, USSR Comput. Math. Math. Phys., 15 (1975), 131-143.  doi: 10.1016/0041-5553(75)90141-X.

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/s0294-1449(16)30405-x.

[3]

A. V. BobylevP. DukesR. Illner and H. D. Victory Jr., On Vlasov-Manev equations. Ⅰ. Foundations, properties, and nonglobal existence, J. Statist. Phys., 88 (1997), 885-911.  doi: 10.1023/B:JOSS.0000015177.60491.3c.

[4]

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.

[5]

F. Bouchut and J. Dolbeault, On long time 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. 

[6]

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.

[7]

J. A. Carrillo, Y.-P. Choi and S. Salem, Propagation of chaos for the Vlasov-Poisson-Fokker-Planck equation with a polynomial cut-off, Commun. Contemp. Math., 21 (2019), 28 pp. doi: 10.1142/S0219199718500396.

[8]

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.

[9]

J. A. CarrilloJ. Soler and J. L. Vázquez, Asymptotic behaviour for the frictionless Vlasov-Poisson-Fokker-Planck system, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1195-1200. 

[10]

Y.-P. Choi and I.-J. Jeong, Well-posedness and singularity formation for Vlasov-Riesz system, preprint, 2022, arXiv: 2201.12988.

[11]

P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions, Ann. Sci. École Norm. Sup. (4), 19 (1986), 519–542. doi: 10.24033/asens.1516.

[12]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[13]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791-817.  doi: 10.1016/S0021-7824(99)00021-5.

[14]

E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. Ⅰ. General theory, Math. Methods Appl. Sci., 3 (1981), 229-248.  doi: 10.1002/mma.1670030117.

[15]

E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. Ⅱ. Special cases, Math. Methods Appl. Sci., 4 (1982), 19-32.  doi: 10.1002/mma.1670040104.

[16]

E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279.  doi: 10.1002/mma.1670060118.

[17]

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.

[18]

R. IllnerH. D. VictoryP. Dukes and A. V. Bobylev, On Vlasov-Manev equations. Ⅱ. Local existence and uniqueness, J. Statist. Phys., 91 (1998), 625-654.  doi: 10.1023/A:1023029711405.

[19]

T. K. 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.

[20]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.

[21]

G. Maneff, Die gravitation und das prinzip von wirkung und gegenwirkung, Z. Physik, 31 (1925), 786-802.  doi: 10.1007/BF02980633.

[22]

G. Maneff, La gravitation et l'énergie au zéro, Comptes Rendues, 190 (1930), 1374-1377. 

[23]

G. Maneff, La gravitation et le principle de l'égalité de l'action et de la réaction, Comptes Rendues, 178 (1924), 2159-2161. 

[24]

G. Maneff, Le principe de la moindre action et la gravitation, Comptes Rendues, 190 (1930), 963-965. 

[25]

C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Comm. Partial Differential Equations, 37 (2012), 1273-1285.  doi: 10.1080/03605302.2011.606863.

[26]

B. Perthame and P. E. Souganidis, A limiting case for velocity averaging, Ann. Sci. École Norm. Sup. (4), 31 (1998), 591–598. doi: 10.1016/S0012-9593(98)80108-0.

[27]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[28]

S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka Math. J., 15 (1978), 245-261. 

[29]

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.

[30]

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.

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