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:
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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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[8]

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

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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.Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

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