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Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows
1. | Department of Mathematics, Florida International University, Miami, FL, 33199 |
2. | Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States |
References:
[1] |
H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.
doi: 10.1007/s00205-008-0160-2. |
[2] |
H. Abels, Longtime behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Nonlocal and abstract parabolic equations and their applications, 9-19, Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, 2009.
doi: 10.4064/bc86-0-1. |
[3] |
H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.
doi: 10.1512/iumj.2008.57.3391. |
[4] |
C. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.
doi: 10.1088/0951-7715/25/11/3211. |
[5] |
T. Caraballo, A. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D LANS-$\alpha$ model, Appl. Math. Optim., 53 (2006), 141-161.
doi: 10.1007/s00245-005-0839-9. |
[6] |
T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin. Dyn. Syst., 4 (2006), 559-578.
doi: 10.3934/dcds.2006.15.559. |
[7] |
T. Caraballo, J. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional lagrange averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 459-479 .
doi: 10.1098/rspa.2005.1574. |
[8] |
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), 5338-5341.
doi: 10.1103/PhysRevLett.81.5338. |
[9] |
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.
doi: 10.1016/S0167-2789(99)00098-6. |
[10] |
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353.
doi: 10.1063/1.870096. |
[11] |
S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83.
doi: 10.1016/S0167-2789(99)00099-8. |
[12] |
L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
[13] |
A. Cheskidov, Turbulent boundary layer equations, C. R. Acad. Sci. Paris Sér. I, 334 (2002), 423-427.
doi: 10.1016/S1631-073X(02)02275-6. |
[14] |
G. Deugoue, Approximation of the trajectory attractor of the 3D MHD system, Comm. Pure Appl. Math., 12 (2013), 2119-2144.
doi: 10.3934/cpaa.2013.12.2119. |
[15] |
C. Foias, D. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes and turbulence theory, J. Dynam. Diff. Equat., 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[16] |
C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 40136.
doi: 10.1016/j.anihpc.2009.11.013. |
[17] |
C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39.
doi: 10.3934/dcds.2010.28.1. |
[18] |
C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 65578.
doi: 10.1007/s11401-010-0603-6. |
[19] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
[20] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. |
[21] |
D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr., 33 (2003), 2355-2365. |
[22] |
J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS- $\alpha$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468.
doi: 10.1098/rsta.2001.0852. |
[23] |
T. Tachim Medjo, Longtime behavior of a 3D LANS-$\alpha$ system with phase transition, Submitted, 2013. |
[24] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[25] |
M. I. Vishik, E. S. Titi and V. V. Chepyzhov, On convergence of solution of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52. |
[26] |
M. I. Vishik, E. S. Titi and V. V. Chepyzhov, On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches 0, Mat. Sb., 198 (2007), 3-36.
doi: 10.1070/SM2007v198n12ABEH003902. |
show all references
References:
[1] |
H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.
doi: 10.1007/s00205-008-0160-2. |
[2] |
H. Abels, Longtime behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Nonlocal and abstract parabolic equations and their applications, 9-19, Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, 2009.
doi: 10.4064/bc86-0-1. |
[3] |
H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.
doi: 10.1512/iumj.2008.57.3391. |
[4] |
C. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.
doi: 10.1088/0951-7715/25/11/3211. |
[5] |
T. Caraballo, A. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D LANS-$\alpha$ model, Appl. Math. Optim., 53 (2006), 141-161.
doi: 10.1007/s00245-005-0839-9. |
[6] |
T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin. Dyn. Syst., 4 (2006), 559-578.
doi: 10.3934/dcds.2006.15.559. |
[7] |
T. Caraballo, J. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional lagrange averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 459-479 .
doi: 10.1098/rspa.2005.1574. |
[8] |
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), 5338-5341.
doi: 10.1103/PhysRevLett.81.5338. |
[9] |
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.
doi: 10.1016/S0167-2789(99)00098-6. |
[10] |
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353.
doi: 10.1063/1.870096. |
[11] |
S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83.
doi: 10.1016/S0167-2789(99)00099-8. |
[12] |
L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
[13] |
A. Cheskidov, Turbulent boundary layer equations, C. R. Acad. Sci. Paris Sér. I, 334 (2002), 423-427.
doi: 10.1016/S1631-073X(02)02275-6. |
[14] |
G. Deugoue, Approximation of the trajectory attractor of the 3D MHD system, Comm. Pure Appl. Math., 12 (2013), 2119-2144.
doi: 10.3934/cpaa.2013.12.2119. |
[15] |
C. Foias, D. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes and turbulence theory, J. Dynam. Diff. Equat., 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[16] |
C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 40136.
doi: 10.1016/j.anihpc.2009.11.013. |
[17] |
C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39.
doi: 10.3934/dcds.2010.28.1. |
[18] |
C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 65578.
doi: 10.1007/s11401-010-0603-6. |
[19] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
[20] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. |
[21] |
D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr., 33 (2003), 2355-2365. |
[22] |
J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS- $\alpha$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468.
doi: 10.1098/rsta.2001.0852. |
[23] |
T. Tachim Medjo, Longtime behavior of a 3D LANS-$\alpha$ system with phase transition, Submitted, 2013. |
[24] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[25] |
M. I. Vishik, E. S. Titi and V. V. Chepyzhov, On convergence of solution of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52. |
[26] |
M. I. Vishik, E. S. Titi and V. V. Chepyzhov, On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches 0, Mat. Sb., 198 (2007), 3-36.
doi: 10.1070/SM2007v198n12ABEH003902. |
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