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

Previous Article
Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case
CPAA Home
This Issue
Next Article
Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping

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

Abstract
Related Papers

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 and 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, 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.

[1]	Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
[2]	T. Tachim Medjo. Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281
[3]	Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009
[4]	T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure and Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[5]	Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
[6]	Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure and Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303
[7]	Andrea Giorgini, Roger Temam. Attractors for the Navier-Stokes-Cahn-Hilliard system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2249-2274. doi: 10.3934/dcdss.2022118
[8]	Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[9]	Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
[10]	Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481
[11]	Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19
[12]	Aseel Farhat, M. S Jolly, Evelyn Lunasin. Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2127-2140. doi: 10.3934/cpaa.2014.13.2127
[13]	Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[14]	Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[15]	Bo You. Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021251
[16]	Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, Andrey Yu. Goritsky. Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2375-2393. doi: 10.3934/dcds.2017103
[17]	Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947
[18]	Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391
[19]	Maicon Sônego, Arnaldo Simal do Nascimento. Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3297-3311. doi: 10.3934/dcdsb.2021185
[20]	Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Ciprian G. Gal T. Tachim Medjo
on Google Scholar
Ciprian G. Gal T. Tachim Medjo

[Back to Top]