American Institute of Mathematical Sciences

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March  2011, 4(1): 227-258. doi: 10.3934/krm.2011.4.227

Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system

José A. Carrillo 1, , Renjun Duan 2, and  Ayman Moussa 3,

1. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
2. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
3. 

UPMC Université Paris 6, UMR 7598 LJLL, F-75005, Paris, France

Received  June 2010 Revised  December 2010 Published  January 2011

We are concerned with the global well-posedness of a two-phase flow system arising in the modelling of fluid-particle interactions. This system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the incompressible Euler equations for a dense phase (fluid) through the friction forcing. Global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state, and moreover an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the macro-micro decomposition and Kawashima's hyperbolic-parabolic dissipation argument. This result is generalized to the periodic case, when particles are in the torus, improving the rate of convergence to exponential.
Keywords: rate of convergence., global well-posedness, Two-phase flow system.
Mathematics Subject Classification: Primary: 35Q84, 35Q31; Secondary: 35B3.
Citation: José A. Carrillo, Renjun Duan, Ayman Moussa. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinetic & Related Models, 2011, 4 (1) : 227-258. doi: 10.3934/krm.2011.4.227
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.  Google Scholar

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C. Baranger, L. Boudin, P.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, C{EMRACS} 2004--Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47.  Google Scholar

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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

[5]

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

[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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

R.-J. Duan, Stability of the Boltzmann equation with potential forces on Torus, Phys. D, 238 (2009), 1808-1820. doi: 10.1016/j.physd.2009.06.007.  Google Scholar

[11]

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.  Google Scholar

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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.  Google Scholar

[13]

T. Goudon, L. He, A. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2009), 2177-2202. doi: 10.1137/090776755.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

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

[18]

S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations Of Magnetohydrodynamics,'' Ph.D thesis, Kyoto University, 1983. Google Scholar

[19]

F. H. Lin, C. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866. doi: 10.1002/cpa.20159.  Google Scholar

[20]

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

[21]

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

[22]

A. Moussa, "Étude Mathématique et Numérique du Transport D'aérosols dans le Poumon Humain,'' Ph.D thesis, ENS Cachan, 2009. Google Scholar

[23]

B. P. O'Dwyer and H. D. Victory, On classical solutions of Vlasov-Poisson-Fokker-Planck systems, Indiana Univ. Math. J., 39 (1990), 105-156. doi: 10.1512/iumj.1990.39.39009.  Google Scholar

[24]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061.  Google Scholar

[25]

C. Sparber, J. A. Carrillo, J. 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

[26]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970.  Google Scholar

[27]

M. E. Taylor, "Partial Differential Equations: Nonlinear Equations," Springer Verlag, 1996.  Google Scholar

[28]

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), 41-80. 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'', (eds. S. Cordier, Th. Goudon, M. Gutnic and E. Sonnendrucker Eds.) IRMA Lect. Math. Theor. Phys. (EMS Publ. House), 7 (2005), 149-176.  Google Scholar

[3]

C. Baranger, L. Boudin, P.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, C{EMRACS} 2004--Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47.  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), 1-26. 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 Integral Equations, 22 (2009), 1247-1271.  Google Scholar

[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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

R.-J. Duan, Stability of the Boltzmann equation with potential forces on Torus, Phys. D, 238 (2009), 1808-1820. doi: 10.1016/j.physd.2009.06.007.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

T. Goudon, L. He, A. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2009), 2177-2202. doi: 10.1137/090776755.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

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

[18]

S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations Of Magnetohydrodynamics,'' Ph.D thesis, Kyoto University, 1983. Google Scholar

[19]

F. H. Lin, C. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866. doi: 10.1002/cpa.20159.  Google Scholar

[20]

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

[21]

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

[22]

A. Moussa, "Étude Mathématique et Numérique du Transport D'aérosols dans le Poumon Humain,'' Ph.D thesis, ENS Cachan, 2009. Google Scholar

[23]

B. P. O'Dwyer and H. D. Victory, On classical solutions of Vlasov-Poisson-Fokker-Planck systems, Indiana Univ. Math. J., 39 (1990), 105-156. doi: 10.1512/iumj.1990.39.39009.  Google Scholar

[24]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061.  Google Scholar

[25]

C. Sparber, J. A. Carrillo, J. 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

[26]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970.  Google Scholar

[27]

M. E. Taylor, "Partial Differential Equations: Nonlinear Equations," Springer Verlag, 1996.  Google Scholar

[28]

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

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