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On a charge interacting with a plasma of unbounded mass
Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system
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 |
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. |
[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.
|
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.
doi: 10.1512/iumj.2004.53.2574. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.
doi: 10.1137/0521061. |
[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. |
[26] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970).
|
[27] |
M. E. Taylor, "Partial Differential Equations: Nonlinear Equations,", Springer Verlag, (1996).
|
[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. |
[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.
|
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.
doi: 10.1512/iumj.2004.53.2574. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.
doi: 10.1137/0521061. |
[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. |
[26] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970).
|
[27] |
M. E. Taylor, "Partial Differential Equations: Nonlinear Equations,", Springer Verlag, (1996).
|
[28] |
F. A. Williams, "Combustion Theory,'', Benjamin Cummings, (1985). Google Scholar |
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