American Institute of Mathematical Sciences

Advanced
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.  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, 7 (2005), 149.   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, 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 Integral 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 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

[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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.  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.  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.  doi: 10.1142/S0218202507002194.  Google Scholar

[22]

A. Moussa, "Étude Mathématique et Numérique du Transport D'aérosols dans le Poumon Humain,'', Ph.D thesis, (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.  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.  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.  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.  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, 7 (2005), 149.   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, 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 Integral 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 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

[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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.  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.  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.  doi: 10.1142/S0218202507002194.  Google Scholar

[22]

A. Moussa, "Étude Mathématique et Numérique du Transport D'aérosols dans le Poumon Humain,'', Ph.D thesis, (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.  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.  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.  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

[1]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
[2]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[3]

Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evolution Equations & Control Theory, 2012, 1 (1) : 171-194. doi: 10.3934/eect.2012.1.171
[4]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021
[5]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198
[6]

Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016
[7]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[8]

Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations & Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011
[9]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[10]

Juliana Honda Lopes, Gabriela Planas. Well-posedness for a non-isothermal flow of two viscous incompressible fluids. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2455-2477. doi: 10.3934/cpaa.2018117
[11]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[12]

Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065
[13]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
[14]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[15]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699
[16]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056
[17]

Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1613-1650. doi: 10.3934/dcds.2019072
[18]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108
[19]

Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Global well-posedness and long time behaviors of chemotaxis-fluid system modeling coral fertilization. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2135-2163. doi: 10.3934/dcds.2020109
[20]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences