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April  2018, 38(4): 2141-2169. doi: 10.3934/dcds.2018088

Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model

Theodore Tachim Medjo

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, USA

Received  May 2017 Revised  October 2017 Published  January 2018

The existence and final fractal dimension of a pullback attractor in the space $ \mathbb{V}$ for a three dimensional system of a non-autonomous globally modified two phase flow on a bounded domain is established under appropriate properties on the time depending forcing term. The model consists of the globally modified Navier-Stokes equations proposed in [6] for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. The existence of the pullback attractors is obtained using the flattening property. Furthermore, we prove that the fractal dimension in $ \mathbb{V}$ of the pullback attractor is finite.

Keywords: Allen-Cahn-Navier-Stokes, globally modified, strong solutions, pullback attractor.
Mathematics Subject Classification: 35Q30, 35Q35, 35Q72.
Citation: Theodore Tachim Medjo. Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088
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]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.

[3]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.

[4]

T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, In A. Johann, H. P. Kruse and F. Rupp, editors, Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics and Statistics, 35 (2013), 473-492.

[5]

T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761.

[6]

T. Caraballo, J. Real and P.E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.

[7]

T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883. doi: 10.1142/S0218127410027428.

[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]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.

[13]

G. Deugoue and J.K. Djoko, On the time discretization for the globally modified three dimensional Navier-Stokes equations, J. Comput. Appl. Math, 235 (2011), 2015-2029. doi: 10.1016/j.cam.2010.10.003.

[14]

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.

[15]

C. 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.

[16]

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.

[17]

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.

[18]

P.C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.

[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. doi: 10.1103/PhysRevLett.80.4173.

[21]

P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[22]

P.E. Kloeden, J.A. Langa and J. Real, Pullback ${V-}$attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.

[23]

P.E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.

[24]

P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. doi: 10.1098/rspa.2007.1831.

[25]

G. Lukaszewicz, On pullback attractors in $ {H}^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258.

[26]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.

[27]

P. Marín-Rubio, J. Real and A.M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.

[28]

A.M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{e}$MA, 51 (2010), 117-124.

[29]

J.E. Marsden and S. Shkoller, Global well-posedness for the {L}agrangian averaged Navier-Stokes (LANS-$ α$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852.

[30]

A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157.

[31]

M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.

[32]

H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $ {H}^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034.

[33]

X.L. Song and Y. Hou, Pullback $\underset{\scriptscriptstyle\centerdot}{-}$attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009.

[34]

T. Tachim Medjo, Unique strong and $ {V-}$attractor of a three dimensional globally modified Allen-Cahn-Navier-Stokes model, Appl. Anal., 96 (2017), 2695-2716. doi: 10.1080/00036811.2016.1236924.

[35]

T. Tachim Medjo, Unique strong and $ {V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2016.1236924, 2016.
[36]

T. Tachim Medjo, Pullback $ {V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2017.1296952, 2017.
[37]

R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997.

[38]

M.I. Vishik, A.I. Komech and A.V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210.

[39]

B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128. doi: 10.1016/j.na.2014.08.018.

[40]

C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous incompressible non-Newtonian fluid, Appl. Math. Comput., 201 (2008), 688-700. doi: 10.1016/j.amc.2008.01.005.

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]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.

[3]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.

[4]

T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, In A. Johann, H. P. Kruse and F. Rupp, editors, Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics and Statistics, 35 (2013), 473-492.

[5]

T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761.

[6]

T. Caraballo, J. Real and P.E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.

[7]

T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883. doi: 10.1142/S0218127410027428.

[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]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.

[13]

G. Deugoue and J.K. Djoko, On the time discretization for the globally modified three dimensional Navier-Stokes equations, J. Comput. Appl. Math, 235 (2011), 2015-2029. doi: 10.1016/j.cam.2010.10.003.

[14]

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.

[15]

C. 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.

[16]

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.

[17]

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.

[18]

P.C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.

[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. doi: 10.1103/PhysRevLett.80.4173.

[21]

P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[22]

P.E. Kloeden, J.A. Langa and J. Real, Pullback ${V-}$attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.

[23]

P.E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.

[24]

P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. doi: 10.1098/rspa.2007.1831.

[25]

G. Lukaszewicz, On pullback attractors in $ {H}^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258.

[26]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.

[27]

P. Marín-Rubio, J. Real and A.M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.

[28]

A.M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{e}$MA, 51 (2010), 117-124.

[29]

J.E. Marsden and S. Shkoller, Global well-posedness for the {L}agrangian averaged Navier-Stokes (LANS-$ α$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852.

[30]

A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157.

[31]

M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.

[32]

H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $ {H}^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034.

[33]

X.L. Song and Y. Hou, Pullback $\underset{\scriptscriptstyle\centerdot}{-}$attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009.

[34]

T. Tachim Medjo, Unique strong and $ {V-}$attractor of a three dimensional globally modified Allen-Cahn-Navier-Stokes model, Appl. Anal., 96 (2017), 2695-2716. doi: 10.1080/00036811.2016.1236924.

[35]

T. Tachim Medjo, Unique strong and $ {V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2016.1236924, 2016.
[36]

T. Tachim Medjo, Pullback $ {V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2017.1296952, 2017.
[37]

R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997.

[38]

M.I. Vishik, A.I. Komech and A.V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210.

[39]

B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128. doi: 10.1016/j.na.2014.08.018.

[40]

C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous incompressible non-Newtonian fluid, Appl. Math. Comput., 201 (2008), 688-700. doi: 10.1016/j.amc.2008.01.005.

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Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761
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P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937
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P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785
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