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 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.

[1]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[2]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

[3]

Hongjie Dong, Yan Guo, Timur Yastrzhembskiy. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition. Kinetic and Related Models, 2022, 15 (3) : 467-516. doi: 10.3934/krm.2022003

[4]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[5]

Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041

[6]

Luis Almeida, Federica Bubba, Benoît Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks and Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934/nhm.2019002

[7]

Ruiying Wei, Yin Li, Zheng-an Yao. Global existence and convergence rates of solutions for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2949-2967. doi: 10.3934/dcdsb.2020047

[8]

Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022009

[9]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[10]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016

[11]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485

[12]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[13]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[14]

Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1409-1415. doi: 10.3934/dcdss.2013.6.1409

[15]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[16]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011

[17]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008

[18]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845

[19]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028

[20]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (157)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]