# American Institute of Mathematical Sciences

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

 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.
Citation: Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687
##### References:
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Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations,, Differential and Integal Equations, 22 (2009), 1247. Google Scholar [6] R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects,, SIAM J. Appl. Math., 43 (1983), 885. doi: 10.1137/0143057. Google Scholar [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. doi: 10.3934/krm.2011.4.227. Google Scholar [8] J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model,, Comm. Partial Differential Equations, 31 (2006), 1349. doi: 10.1080/03605300500394389. Google Scholar [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. doi: 10.1016/j.jde.2011.07.016. Google Scholar [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. doi: 10.3934/dcdsb.2002.2.591. Google Scholar [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. doi: 10.1080/03605309908821418. Google Scholar [12] R.-J. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusions,, Comm. Math. Phys., 300 (2010), 95. doi: 10.1007/s00220-010-1110-z. Google Scholar [13] T. Goudon, Asymptotic problems for a kinetic model of two-phase flow,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1371. doi: 10.1017/S030821050000144X. Google Scholar [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. doi: 10.1137/090776755. Google Scholar [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. doi: 10.1512/iumj.2004.53.2508. Google Scholar [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. doi: 10.1512/iumj.2004.53.2509. Google Scholar [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. doi: 10.4310/CMS.2012.v10.n1.a15. Google Scholar [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. doi: 10.1137/120861515. Google Scholar [19] Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081. doi: 10.1512/iumj.2004.53.2574. 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. doi: 10.1007/BF03167396. Google Scholar [21] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,, Thesis, (1983). Google Scholar [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. doi: 10.1007/s00220-008-0523-4. Google Scholar [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. doi: 10.1142/S0218202507002194. Google Scholar [24] A. Moussa and F. Sueur, On a Vlasov-Euler system for 2D sprays with gyroscopic effects,, Asymptotic Analysis, 81 (2013), 53. doi: 10.3233/ASY-2012-1123. Google Scholar [25] F. A. Williams, Combustion Theory,, Benjamin Cummings, (1985). Google Scholar

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. doi: 10.1137/S0036139902408163. Google Scholar [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, 7 (2005), 149. doi: 10.4171/012-1/8. Google Scholar [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, 14 (2005), 41. Google Scholar [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. doi: 10.1142/S0219891606000707. Google Scholar [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. Google Scholar [6] R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects,, SIAM J. Appl. Math., 43 (1983), 885. doi: 10.1137/0143057. Google Scholar [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. doi: 10.3934/krm.2011.4.227. Google Scholar [8] J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model,, Comm. Partial Differential Equations, 31 (2006), 1349. doi: 10.1080/03605300500394389. Google Scholar [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. doi: 10.1016/j.jde.2011.07.016. Google Scholar [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. doi: 10.3934/dcdsb.2002.2.591. Google Scholar [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. doi: 10.1080/03605309908821418. Google Scholar [12] R.-J. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusions,, Comm. Math. Phys., 300 (2010), 95. doi: 10.1007/s00220-010-1110-z. Google Scholar [13] T. Goudon, Asymptotic problems for a kinetic model of two-phase flow,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1371. doi: 10.1017/S030821050000144X. Google Scholar [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. doi: 10.1137/090776755. Google Scholar [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. doi: 10.1512/iumj.2004.53.2508. Google Scholar [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. doi: 10.1512/iumj.2004.53.2509. Google Scholar [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. doi: 10.4310/CMS.2012.v10.n1.a15. Google Scholar [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. doi: 10.1137/120861515. Google Scholar [19] Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081. doi: 10.1512/iumj.2004.53.2574. 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. doi: 10.1007/BF03167396. Google Scholar [21] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,, Thesis, (1983). Google Scholar [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. doi: 10.1007/s00220-008-0523-4. Google Scholar [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. doi: 10.1142/S0218202507002194. Google Scholar [24] A. Moussa and F. Sueur, On a Vlasov-Euler system for 2D sprays with gyroscopic effects,, Asymptotic Analysis, 81 (2013), 53. doi: 10.3233/ASY-2012-1123. Google Scholar [25] F. A. Williams, Combustion Theory,, Benjamin Cummings, (1985). Google Scholar
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