April 2019, 12(2): 357-396. doi: 10.3934/krm.2019016

A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data

1. 

School of Mathematics, Hefei University of Technology, Hefei 230009, China

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Lan Zhang

Received  March 2018 Revised  July 2018 Published  November 2018

Fund Project: This work was supported by a Grant from National Natural Science Foundation of China under Contract 11671309 and "The Fundamental Research Funds for the Central Universities"

We present a two-dimensional coupled system for particles and compressible conducting fluid in an electromagnetic field interactions, which the kinetic Vlasov-Fokker-Planck model for particle part and the isentropic compressible MHD equations for the fluid part, respectively, and these separate systems are coupled with the drag force. For this specific coupled system, a sufficient framework for the global existence of classical solutions with large initial data which may contain vacuum is established.

Citation: Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016
References:
[1]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differential Equations, 257 (2014), 2225-2255. doi: 10.1016/j.jde.2014.05.035.

[2]

C. BarangerL. BoudinP.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47.

[3]

C. Baranger and L. Desvillettes, Coupling Euler and Vlasov equations in the context of sprays: The local-in-time, classical solutions, J. Hyperbolic Differ. Equ., 3 (2006), 1-26. doi: 10.1142/S0219891606000707.

[4]

S. BerresR. BurgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80. doi: 10.1137/S0036139902408163.

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L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differ. Int. Equations, 22 (2009), 1247-1271.

[6]

L. BoudinL. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657-669. doi: 10.4310/CMS.2003.v1.n4.a2.

[7]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[8]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259-275.

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J. A. CarrilloR.-J. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system, Kinet. Relat. Models., 4 (2011), 227-258. doi: 10.3934/krm.2011.4.227.

[10]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and non-existence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[11]

R. CoifmanR. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. doi: 10.2307/1970954.

[12]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331. doi: 10.1090/S0002-9947-1975-0380244-8.

[13]

R.-J. Duan and S.-Q. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinet. Relat. Models, 6 (2013), 687-700. doi: 10.3934/krm.2013.6.687.

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H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541-544.

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E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press Inc., 2004.

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D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.

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T. GoudonL. HeA. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202. doi: 10.1137/090776755.

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S.-Y. Ha, B.-K. Huang, Q.-H. Xiao and X.-T. Zhang, A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data, Submitted.

[19]

S.-Y. HaB.-K. HuangQ.-H. Xiao and X.-T. Zhang, Global classical solutions to 1D coupled system of flocking particles and compressible fluids with large initial data, Math. Models Methods Appl. Sci., 28 (2018), 1-60. doi: 10.1142/S021820251850001X.

[20]

K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74. doi: 10.1007/BF03167396.

[21]

X.-P. Hu and D.-H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.

[22]

X.-P. Hu and D.-H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.

[23]

X.-D. Huang and J. Li, Global well-posedness of classical solutions to the Cauchy problem of two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, arXiv: 1207.3746v1.

[24]

X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123-154. doi: 10.1016/j.matpur.2016.02.003.

[25]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521. doi: 10.1007/s00021-014-0171-8.

[26]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014.

[27]

F.-C. LiY.-M. Mu and D.-H. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026. doi: 10.1137/15M1053049.

[28]

P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1. Incompressible models, Oxford University Press, New York, 1996.

[29]

P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, Oxford University Press, New York, 1998.

[30]

Y. Mei, Global classical solutions to the 2D compressible MHD equations with large data and vacuum, J. Differential Equations, 258 (2015), 3304-3359. doi: 10.1016/j.jde.2014.11.023.

[31]

Y. Mei, Corrigendum to "Global classical solutions to the 2D compressible MHD equations with large data and vacuum", J. Differential Equations, 258 (2015), 3360-3362. doi: 10.1016/j.jde.2015.02.001.

[32]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596. doi: 10.1007/s00220-008-0523-4.

[33]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.

[34]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.

[35]

C. SparberJ.-A. CarrilloJ. Dolbeault and P.-A. Markowich, On the long-time behavior of the quantum Fokker-Planck equation, Monatsh. Math., 141 (2004), 237-257. doi: 10.1007/s00605-003-0043-4.

[36]

F.-A. Williams, Combustion Theory, Benjamin Cummings, 1985.

[37]

V.-A. Vaigant and A.-V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316. doi: 10.1007/BF02106835.

show all references

References:
[1]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differential Equations, 257 (2014), 2225-2255. doi: 10.1016/j.jde.2014.05.035.

[2]

C. BarangerL. BoudinP.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47.

[3]

C. Baranger and L. Desvillettes, Coupling Euler and Vlasov equations in the context of sprays: The local-in-time, classical solutions, J. Hyperbolic Differ. Equ., 3 (2006), 1-26. doi: 10.1142/S0219891606000707.

[4]

S. BerresR. BurgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80. doi: 10.1137/S0036139902408163.

[5]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differ. Int. Equations, 22 (2009), 1247-1271.

[6]

L. BoudinL. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657-669. doi: 10.4310/CMS.2003.v1.n4.a2.

[7]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[8]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259-275.

[9]

J. A. CarrilloR.-J. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system, Kinet. Relat. Models., 4 (2011), 227-258. doi: 10.3934/krm.2011.4.227.

[10]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and non-existence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[11]

R. CoifmanR. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. doi: 10.2307/1970954.

[12]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331. doi: 10.1090/S0002-9947-1975-0380244-8.

[13]

R.-J. Duan and S.-Q. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinet. Relat. Models, 6 (2013), 687-700. doi: 10.3934/krm.2013.6.687.

[14]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541-544.

[15]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press Inc., 2004.

[16]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.

[17]

T. GoudonL. HeA. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202. doi: 10.1137/090776755.

[18]

S.-Y. Ha, B.-K. Huang, Q.-H. Xiao and X.-T. Zhang, A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data, Submitted.

[19]

S.-Y. HaB.-K. HuangQ.-H. Xiao and X.-T. Zhang, Global classical solutions to 1D coupled system of flocking particles and compressible fluids with large initial data, Math. Models Methods Appl. Sci., 28 (2018), 1-60. doi: 10.1142/S021820251850001X.

[20]

K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74. doi: 10.1007/BF03167396.

[21]

X.-P. Hu and D.-H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.

[22]

X.-P. Hu and D.-H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.

[23]

X.-D. Huang and J. Li, Global well-posedness of classical solutions to the Cauchy problem of two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, arXiv: 1207.3746v1.

[24]

X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123-154. doi: 10.1016/j.matpur.2016.02.003.

[25]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521. doi: 10.1007/s00021-014-0171-8.

[26]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014.

[27]

F.-C. LiY.-M. Mu and D.-H. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026. doi: 10.1137/15M1053049.

[28]

P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1. Incompressible models, Oxford University Press, New York, 1996.

[29]

P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, Oxford University Press, New York, 1998.

[30]

Y. Mei, Global classical solutions to the 2D compressible MHD equations with large data and vacuum, J. Differential Equations, 258 (2015), 3304-3359. doi: 10.1016/j.jde.2014.11.023.

[31]

Y. Mei, Corrigendum to "Global classical solutions to the 2D compressible MHD equations with large data and vacuum", J. Differential Equations, 258 (2015), 3360-3362. doi: 10.1016/j.jde.2015.02.001.

[32]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596. doi: 10.1007/s00220-008-0523-4.

[33]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.

[34]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.

[35]

C. SparberJ.-A. CarrilloJ. Dolbeault and P.-A. Markowich, On the long-time behavior of the quantum Fokker-Planck equation, Monatsh. Math., 141 (2004), 237-257. doi: 10.1007/s00605-003-0043-4.

[36]

F.-A. Williams, Combustion Theory, Benjamin Cummings, 1985.

[37]

V.-A. Vaigant and A.-V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316. doi: 10.1007/BF02106835.

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