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Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model
Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, USA |
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 [
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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