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December  2013, 6(4): 687-700. doi: 10.3934/krm.2013.6.687

Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force

Renjun Duan 1, and  Shuangqian Liu 2,

1. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2. 

Department of Mathematics, Jinan Unviersity, Guangdong

Received  March 2013 Revised  June 2013 Published  November 2013

We are concerned with a two-phase flow system consisting of the Vlasov-Fokker-Planck equation for particles coupled to the compressible Euler equations for the fluid through the friction force. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The proof is based on the classical energy estimates.
Keywords: stability, convergence rates., Fokker-Planck equations, Euler equations.
Mathematics Subject Classification: Primary: 35Q84, 35Q31; Secondary: 35B3.
Citation: Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic and Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687
References:
[1]

S. Berres, R. Bürger, K. 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.

[2]

C. Baranger, G. Baudin, L. Boudin, B. Després, F. Lagoutière, E. Lapébie and T. Takahashi, Liquid jet generation and break-up, in Numerical Methods for Hyperbolic and Kinetic Equations, S. Cordier, Th. Goudon, M. Gutnic, E. Sonnendrucker Eds., IRMA Lectures in Mathematics and Theoretical Physics (EMS Publ. House) 7 (2005), 149-176. doi: 10.4171/012-1/8.

[3]

C. Baranger, L. Boudin, P.-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.

[4]

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.

[5]

L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differential and Integal Equations, 22 (2009), 1247-1271.

[6]

R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects, SIAM J. Appl. Math., 43 (1983), 885-906. doi: 10.1137/0143057.

[7]

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

[8]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Comm. Partial Differential Equations, 31 (2006), 1349-1379. doi: 10.1080/03605300500394389.

[9]

M. Chae, K. Kang and J. Lee, Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations, Journal of Differential Equations, 251 (2011), 2431-2465. doi: 10.1016/j.jde.2011.07.016.

[10]

K. Domelevo, Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 591-607. doi: 10.3934/dcdsb.2002.2.591.

[11]

K. Domelevo and J. M. Roquejoffre, Existence and stability of travelling wave solutions in a kinetic model of two-phase flows, Comm. PDE, 24 (1999), 61-108. doi: 10.1080/03605309908821418.

[12]

R.-J. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusions, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[13]

T. Goudon, Asymptotic problems for a kinetic model of two-phase flow, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1371-1384. doi: 10.1017/S030821050000144X.

[14]

T. Goudon, L. He, A. 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.

[15]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508.

[16]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.

[17]

T. Goudon, S. Jin and B. Yan, Simulation of fluid-particles flows: Heavy particles, flowing regime, and asymptotic-preserving schemes, Commun. Math. Sci., 10 (2012), 355-385. doi: 10.4310/CMS.2012.v10.n1.a15.

[18]

T. Goudon, M. Sy and L. Tiné, A fluid-kinetic model for particulate flows with coagulation and breakup: Stationary solutions, stability, and hydrodynamic regimes, SIAM Journal on Applied Mathematics, 73 (2013), 401-421. doi: 10.1137/120861515.

[19]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.

[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]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Thesis, Kyoto University, 1983.

[22]

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.

[23]

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.

[24]

A. Moussa and F. Sueur, On a Vlasov-Euler system for 2D sprays with gyroscopic effects, Asymptotic Analysis, 81 (2013), 53-91. doi: 10.3233/ASY-2012-1123.

[25]

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

show all references

References:
[1]

S. Berres, R. Bürger, K. 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.

[2]

C. Baranger, G. Baudin, L. Boudin, B. Després, F. Lagoutière, E. Lapébie and T. Takahashi, Liquid jet generation and break-up, in Numerical Methods for Hyperbolic and Kinetic Equations, S. Cordier, Th. Goudon, M. Gutnic, E. Sonnendrucker Eds., IRMA Lectures in Mathematics and Theoretical Physics (EMS Publ. House) 7 (2005), 149-176. doi: 10.4171/012-1/8.

[3]

C. Baranger, L. Boudin, P.-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.

[4]

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.

[5]

L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differential and Integal Equations, 22 (2009), 1247-1271.

[6]

R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects, SIAM J. Appl. Math., 43 (1983), 885-906. doi: 10.1137/0143057.

[7]

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

[8]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Comm. Partial Differential Equations, 31 (2006), 1349-1379. doi: 10.1080/03605300500394389.

[9]

M. Chae, K. Kang and J. Lee, Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations, Journal of Differential Equations, 251 (2011), 2431-2465. doi: 10.1016/j.jde.2011.07.016.

[10]

K. Domelevo, Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 591-607. doi: 10.3934/dcdsb.2002.2.591.

[11]

K. Domelevo and J. M. Roquejoffre, Existence and stability of travelling wave solutions in a kinetic model of two-phase flows, Comm. PDE, 24 (1999), 61-108. doi: 10.1080/03605309908821418.

[12]

R.-J. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusions, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[13]

T. Goudon, Asymptotic problems for a kinetic model of two-phase flow, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1371-1384. doi: 10.1017/S030821050000144X.

[14]

T. Goudon, L. He, A. 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.

[15]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508.

[16]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.

[17]

T. Goudon, S. Jin and B. Yan, Simulation of fluid-particles flows: Heavy particles, flowing regime, and asymptotic-preserving schemes, Commun. Math. Sci., 10 (2012), 355-385. doi: 10.4310/CMS.2012.v10.n1.a15.

[18]

T. Goudon, M. Sy and L. Tiné, A fluid-kinetic model for particulate flows with coagulation and breakup: Stationary solutions, stability, and hydrodynamic regimes, SIAM Journal on Applied Mathematics, 73 (2013), 401-421. doi: 10.1137/120861515.

[19]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.

[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]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Thesis, Kyoto University, 1983.

[22]

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.

[23]

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.

[24]

A. Moussa and F. Sueur, On a Vlasov-Euler system for 2D sprays with gyroscopic effects, Asymptotic Analysis, 81 (2013), 53-91. doi: 10.3233/ASY-2012-1123.

[25]

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

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