-
Previous Article
Unstable galaxy models
- KRM Home
- This Issue
-
Next Article
A Milne problem from a Bose condensate with excitations
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 |
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] |
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] |
[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
Tools
Metrics
Other articles
by authors
[Back to Top]