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November  2014, 13(6): 2229-2252. doi: 10.3934/cpaa.2014.13.2229

Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows

Ciprian G. Gal 1, and  T. Tachim Medjo 2,

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

Received  June 2013 Revised  April 2014 Published  July 2014

In this article we study the relations between the long-time dynamics of the 3D Allen-Cahn-LANS-$\alpha$ model and the exact 3D Allen-Cahn-Navier-Stokes system. Following the idea of [26], we prove that bounded set of solutions of the Allen-Cahn-LANS-$\alpha$ model converge to the trajectory attractor $\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes system as time goes to $+ \infty $ and $\alpha$ approaches $0^+. $ In particular we show that the trajectory attractors $\mathcal{U}_{\alpha} $ of the 3D Allen-Cahn-LANS-$\alpha$ model converges to the trajectory attractor $\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes as $ \alpha $ approaches $0^+. $ Let us mention that the strong nonlinearity that results from the coupling of the convective Allen-Cahn system and the LANS-$\alpha$ equations makes the analysis of the problem considered in this article more involved.
Keywords: Lagrange averaged Navier-Stokes-$\alpha$, trajectory attractors., Allen-Cahn system, phase transition.
Mathematics Subject Classification: 35Q30, 35Q35, 35Q7.
Citation: Ciprian G. Gal, T. Tachim Medjo. Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2229-2252. doi: 10.3934/cpaa.2014.13.2229
References:
[1]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities,, \emph{Arch. Ration. Mech. Anal.}, 194 (2009), 463.  doi: 10.1007/s00205-008-0160-2.  Google Scholar

[2]

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H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids,, \emph{Indiana Univ. Math. J.}, 57 (2008), 659.  doi: 10.1512/iumj.2008.57.3391.  Google Scholar

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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,, \emph{Nonlinearity}, 25 (2012), 3211.  doi: 10.1088/0951-7715/25/11/3211.  Google Scholar

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T. Caraballo, A. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D LANS-$\alpha$ model,, \emph{Appl. Math. Optim.}, 53 (2006), 141.  doi: 10.1007/s00245-005-0839-9.  Google Scholar

[6]

T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay,, \emph{Discrete Contin. Dyn. Syst.}, 4 (2006), 559.  doi: 10.3934/dcds.2006.15.559.  Google Scholar

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T. Caraballo, J. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional lagrange averaged Navier-Stokes equations,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 462 (2006), 459.  doi: 10.1098/rspa.2005.1574.  Google Scholar

[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,, \emph{Phys. Rev. Lett.}, 81 (1998), 5338.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[9]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence,, \emph{Physica D}, 133 (1999), 49.  doi: 10.1016/S0167-2789(99)00098-6.  Google Scholar

[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,, \emph{Phys. Fluids}, 11 (1999), 2343.  doi: 10.1063/1.870096.  Google Scholar

[11]

S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model,, \emph{Physica D}, 133 (1999), 66.  doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar

[12]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, \emph{Milan J. Math.}, 79 (2011), 561.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[13]

A. Cheskidov, Turbulent boundary layer equations,, \emph{C. R. Acad. Sci. Paris S\'er. I}, 334 (2002), 423.  doi: 10.1016/S1631-073X(02)02275-6.  Google Scholar

[14]

G. Deugoue, Approximation of the trajectory attractor of the 3D MHD system,, \emph{Comm. Pure Appl. Math.}, 12 (2013), 2119.  doi: 10.3934/cpaa.2013.12.2119.  Google Scholar

[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,, \emph{J. Dynam. Diff. Equat.}, 14 (2002), 1.  doi: 10.1023/A:1012984210582.  Google Scholar

[16]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010).  doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[17]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1.  doi: 10.3934/dcds.2010.28.1.  Google Scholar

[18]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D,, \emph{Chin. Ann. Math. Ser. B}, 31 (2010).  doi: 10.1007/s11401-010-0603-6.  Google Scholar

[19]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories,, \emph{Adv. Math.}, 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[20]

D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, \emph{Phys. Rev. Lett.}, 349 (1998), 4173.   Google Scholar

[21]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, \emph{J. Phys. Oceanogr.}, 33 (2003), 2355.   Google Scholar

[22]

J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS- $\alpha$) equations on bounded domains,, \emph{Phil. Trans. R. Soc. Lond. A}, 359 (2001), 1449.  doi: 10.1098/rsta.2001.0852.  Google Scholar

[23]

T. Tachim Medjo, Longtime behavior of a 3D LANS-$\alpha$ system with phase transition,, Submitted, (2013).   Google Scholar

[24]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, volume 68, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[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,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 33.   Google Scholar

[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,, \emph{Mat. Sb.}, 198 (2007), 3.  doi: 10.1070/SM2007v198n12ABEH003902.  Google Scholar

show all references

References:
[1]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities,, \emph{Arch. Ration. Mech. Anal.}, 194 (2009), 463.  doi: 10.1007/s00205-008-0160-2.  Google Scholar

[2]

H. Abels, Longtime behavior of solutions of a Navier-Stokes/Cahn-Hilliard system,, \emph{Nonlocal and abstract parabolic equations and their applications}, (2009), 9.  doi: 10.4064/bc86-0-1.  Google Scholar

[3]

H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids,, \emph{Indiana Univ. Math. J.}, 57 (2008), 659.  doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[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,, \emph{Nonlinearity}, 25 (2012), 3211.  doi: 10.1088/0951-7715/25/11/3211.  Google Scholar

[5]

T. Caraballo, A. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D LANS-$\alpha$ model,, \emph{Appl. Math. Optim.}, 53 (2006), 141.  doi: 10.1007/s00245-005-0839-9.  Google Scholar

[6]

T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay,, \emph{Discrete Contin. Dyn. Syst.}, 4 (2006), 559.  doi: 10.3934/dcds.2006.15.559.  Google Scholar

[7]

T. Caraballo, J. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional lagrange averaged Navier-Stokes equations,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 462 (2006), 459.  doi: 10.1098/rspa.2005.1574.  Google Scholar

[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,, \emph{Phys. Rev. Lett.}, 81 (1998), 5338.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[9]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence,, \emph{Physica D}, 133 (1999), 49.  doi: 10.1016/S0167-2789(99)00098-6.  Google Scholar

[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,, \emph{Phys. Fluids}, 11 (1999), 2343.  doi: 10.1063/1.870096.  Google Scholar

[11]

S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model,, \emph{Physica D}, 133 (1999), 66.  doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar

[12]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, \emph{Milan J. Math.}, 79 (2011), 561.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[13]

A. Cheskidov, Turbulent boundary layer equations,, \emph{C. R. Acad. Sci. Paris S\'er. I}, 334 (2002), 423.  doi: 10.1016/S1631-073X(02)02275-6.  Google Scholar

[14]

G. Deugoue, Approximation of the trajectory attractor of the 3D MHD system,, \emph{Comm. Pure Appl. Math.}, 12 (2013), 2119.  doi: 10.3934/cpaa.2013.12.2119.  Google Scholar

[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,, \emph{J. Dynam. Diff. Equat.}, 14 (2002), 1.  doi: 10.1023/A:1012984210582.  Google Scholar

[16]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010).  doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[17]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1.  doi: 10.3934/dcds.2010.28.1.  Google Scholar

[18]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D,, \emph{Chin. Ann. Math. Ser. B}, 31 (2010).  doi: 10.1007/s11401-010-0603-6.  Google Scholar

[19]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories,, \emph{Adv. Math.}, 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[20]

D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, \emph{Phys. Rev. Lett.}, 349 (1998), 4173.   Google Scholar

[21]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, \emph{J. Phys. Oceanogr.}, 33 (2003), 2355.   Google Scholar

[22]

J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS- $\alpha$) equations on bounded domains,, \emph{Phil. Trans. R. Soc. Lond. A}, 359 (2001), 1449.  doi: 10.1098/rsta.2001.0852.  Google Scholar

[23]

T. Tachim Medjo, Longtime behavior of a 3D LANS-$\alpha$ system with phase transition,, Submitted, (2013).   Google Scholar

[24]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, volume 68, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[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,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 33.   Google Scholar

[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,, \emph{Mat. Sb.}, 198 (2007), 3.  doi: 10.1070/SM2007v198n12ABEH003902.  Google Scholar

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