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Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays
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On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities
A Cahn-Hilliard-Navier-Stokes model with delays
1. | Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States |
References:
[1] |
T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123.
doi: 10.1088/0022-3727/32/10/307. |
[2] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[3] |
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., 15 (2006), 559-578.
doi: 10.3934/dcds.2006.15.559. |
[4] |
T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[5] |
T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.
doi: 10.1098/rspa.2003.1166. |
[6] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[7] |
E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160.
doi: 10.1142/S0218202510004544. |
[8] |
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), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[9] |
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. |
[10] |
C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678.
doi: 10.1007/s11401-010-0603-6. |
[11] |
P.C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. |
[12] |
T. Tachim Medjo, Pullback attractors for a non-autonomous homogeneous two-phase flow model, J. Diff. Equa., 253 (2012), 1779-1806.
doi: 10.1016/j.jde.2012.06.004. |
[13] |
A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, (2009), 641-709.
doi: 10.1017/CBO9780511534874.012. |
[14] |
T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018.
doi: 10.3934/dcds.2005.12.997. |
[15] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997.
doi: 10.1007/978-1-4612-0645-3. |
show all references
References:
[1] |
T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123.
doi: 10.1088/0022-3727/32/10/307. |
[2] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[3] |
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., 15 (2006), 559-578.
doi: 10.3934/dcds.2006.15.559. |
[4] |
T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[5] |
T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.
doi: 10.1098/rspa.2003.1166. |
[6] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[7] |
E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160.
doi: 10.1142/S0218202510004544. |
[8] |
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), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[9] |
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. |
[10] |
C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678.
doi: 10.1007/s11401-010-0603-6. |
[11] |
P.C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. |
[12] |
T. Tachim Medjo, Pullback attractors for a non-autonomous homogeneous two-phase flow model, J. Diff. Equa., 253 (2012), 1779-1806.
doi: 10.1016/j.jde.2012.06.004. |
[13] |
A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, (2009), 641-709.
doi: 10.1017/CBO9780511534874.012. |
[14] |
T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018.
doi: 10.3934/dcds.2005.12.997. |
[15] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997.
doi: 10.1007/978-1-4612-0645-3. |
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