-
Previous Article
An application of Moser's twist theorem to superlinear impulsive differential equations
- DCDS Home
- This Issue
-
Next Article
Qualitative properties of positive solutions for mixed integro-differential equations
On the convergence of a stochastic 3D globally modified two-phase flow model
Department of Mathematics and Satistics, Florida International University, MMC, Miami, Florida 33199, USA |
We study in this article a stochastic 3D globally modified Allen-Cahn-Navier-Stokes model in a bounded domain. We prove the existence and uniqueness of a strong solutions. The proof relies on a Galerkin approximation, as well as some compactness results. Furthermore, we discuss the relation between the stochastic 3D globally modified Allen-Cahn-Navier-Stokes equations and the stochastic 3D Allen-Cahn-Navier-Stokes equations, by proving a convergence theorem. More precisely, as a parameter $N$ tends to infinity, a subsequence of solutions of the stochastic 3D globally modified Allen-Cahn-Navier-Stokes equations converges to a weak martingale solution of the stochastic 3D Allen-Cahn-Navier-Stokes equations.
References:
[1] |
A. Bensoussan,
Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.
doi: 10.1007/BF00996149. |
[2] |
T. Blesgen,
A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123.
|
[3] |
D. Breit, E. Feireisl and M. Hofmanová,
Incompressible limit for compressible fluids with stochastic forcing, Arch. Ration. Mech. Anal., 222 (2016), 895-926.
doi: 10.1007/s00205-016-1014-y. |
[4] |
Z. Brzeźiak, W Liu and J. Zhu,
Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.
doi: 10.1016/j.nonrwa.2013.12.005. |
[5] |
Z. Brzeźniak, E. Hausenblas and J. Zhu,
2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.
doi: 10.1016/j.na.2012.10.011. |
[6] |
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. |
[7] |
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.
doi: 10.1515/ans-2006-0304. |
[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] |
A. Debussche, N. Glatt-Holtz and R. Temam,
Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.
doi: 10.1016/j.physd.2011.03.009. |
[13] |
G. Deugoué and T. Tachim Medjo,
The stochastic 3D globally modified Navier-Stokes Equations: Existence, Uniqueness and Asymptotic behavior, Commun. Pure Appl. Anal., 17 (2018), 2593-2621.
doi: 10.3934/cpaa.2018123. |
[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] |
F. Flandoli and D. Gatarek,
Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 307-391.
doi: 10.1007/BF01192467. |
[16] |
F. Flandoli and B. Maslowski,
Ergodicity of the 2D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171 (1995), 119-141.
doi: 10.1007/BF02104513. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
N. Glatz-Holtz and M. Ziane,
Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600.
|
[21] |
P. C. Hohenberg and B. I. Halperin,
Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.
|
[22] |
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. |
[23] |
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.
|
[24] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland, Kodansha, 1989. |
[25] |
J. E. Marsden and S. Shkoller,
Global well-posedness for the Lagrangian 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. |
[26] |
A. Onuki,
Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157.
|
[27] |
G. Da Prato and A. Debussche,
Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947.
doi: 10.1016/S0021-7824(03)00025-4. |
[28] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781107295513. |
[29] |
C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, 2007. |
[30] |
M. R${\ddot o}$ckner and T. Zhang,
Stochastic 3D tamed Navier-Stokes equations: Existence, Uniqueness and small time large deviation principles, J. Differential Equations, 252 (2012), 716-744.
doi: 10.1016/j.jde.2011.09.030. |
[31] |
M. Romito,
The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.
doi: 10.1515/ans-2009-0209. |
[32] |
A. V. Skorohod I. I. Gikhman, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. |
[33] |
T. Tachim Medjo,
A non-autonomous two-phase flow model with oscillating external force and its global attractor, Nonlinear Anal., 75 (2012), 226-243.
doi: 10.1016/j.na.2011.08.024. |
[34] |
T. Tachim Medjo,
Pullback attractors for a non-autonomous homogeneous two-phase flow model, J. Differential Equations, 253 (2012), 1779-1806.
doi: 10.1016/j.jde.2012.06.004. |
[35] |
T. Tachim Medjo,
Unique strong and V-attractor of a three dimensional globally modified two-phase flow model, Ann. Mat. Pura Appl., 197 (2018), 843-868.
doi: 10.1007/s10231-017-0706-8. |
[36] |
T. Tachim Medjo and F. Tone,
Long time stability of a classical efficient scheme for an incompressible two-phase flow model, Asymptot. Anal., 95 (2015), 101-127.
doi: 10.3233/ASY-151325. |
[37] |
R. Temam, Infinite 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. |
[38] |
M. I. Vishik, A. I. Komech and A. V. Fursikov,
Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210,256.
|
show all references
References:
[1] |
A. Bensoussan,
Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.
doi: 10.1007/BF00996149. |
[2] |
T. Blesgen,
A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123.
|
[3] |
D. Breit, E. Feireisl and M. Hofmanová,
Incompressible limit for compressible fluids with stochastic forcing, Arch. Ration. Mech. Anal., 222 (2016), 895-926.
doi: 10.1007/s00205-016-1014-y. |
[4] |
Z. Brzeźiak, W Liu and J. Zhu,
Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.
doi: 10.1016/j.nonrwa.2013.12.005. |
[5] |
Z. Brzeźniak, E. Hausenblas and J. Zhu,
2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.
doi: 10.1016/j.na.2012.10.011. |
[6] |
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. |
[7] |
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.
doi: 10.1515/ans-2006-0304. |
[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] |
A. Debussche, N. Glatt-Holtz and R. Temam,
Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.
doi: 10.1016/j.physd.2011.03.009. |
[13] |
G. Deugoué and T. Tachim Medjo,
The stochastic 3D globally modified Navier-Stokes Equations: Existence, Uniqueness and Asymptotic behavior, Commun. Pure Appl. Anal., 17 (2018), 2593-2621.
doi: 10.3934/cpaa.2018123. |
[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] |
F. Flandoli and D. Gatarek,
Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 307-391.
doi: 10.1007/BF01192467. |
[16] |
F. Flandoli and B. Maslowski,
Ergodicity of the 2D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171 (1995), 119-141.
doi: 10.1007/BF02104513. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
N. Glatz-Holtz and M. Ziane,
Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600.
|
[21] |
P. C. Hohenberg and B. I. Halperin,
Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.
|
[22] |
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. |
[23] |
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.
|
[24] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland, Kodansha, 1989. |
[25] |
J. E. Marsden and S. Shkoller,
Global well-posedness for the Lagrangian 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. |
[26] |
A. Onuki,
Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157.
|
[27] |
G. Da Prato and A. Debussche,
Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947.
doi: 10.1016/S0021-7824(03)00025-4. |
[28] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781107295513. |
[29] |
C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, 2007. |
[30] |
M. R${\ddot o}$ckner and T. Zhang,
Stochastic 3D tamed Navier-Stokes equations: Existence, Uniqueness and small time large deviation principles, J. Differential Equations, 252 (2012), 716-744.
doi: 10.1016/j.jde.2011.09.030. |
[31] |
M. Romito,
The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.
doi: 10.1515/ans-2009-0209. |
[32] |
A. V. Skorohod I. I. Gikhman, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. |
[33] |
T. Tachim Medjo,
A non-autonomous two-phase flow model with oscillating external force and its global attractor, Nonlinear Anal., 75 (2012), 226-243.
doi: 10.1016/j.na.2011.08.024. |
[34] |
T. Tachim Medjo,
Pullback attractors for a non-autonomous homogeneous two-phase flow model, J. Differential Equations, 253 (2012), 1779-1806.
doi: 10.1016/j.jde.2012.06.004. |
[35] |
T. Tachim Medjo,
Unique strong and V-attractor of a three dimensional globally modified two-phase flow model, Ann. Mat. Pura Appl., 197 (2018), 843-868.
doi: 10.1007/s10231-017-0706-8. |
[36] |
T. Tachim Medjo and F. Tone,
Long time stability of a classical efficient scheme for an incompressible two-phase flow model, Asymptot. Anal., 95 (2015), 101-127.
doi: 10.3233/ASY-151325. |
[37] |
R. Temam, Infinite 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. |
[38] |
M. I. Vishik, A. I. Komech and A. V. Fursikov,
Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210,256.
|
[1] |
T. Tachim Medjo. On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021282 |
[2] |
G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123 |
[3] |
Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761 |
[4] |
Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779 |
[5] |
Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211 |
[6] |
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 and Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785 |
[7] |
P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure and Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937 |
[8] |
Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655 |
[9] |
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 |
[10] |
Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 |
[11] |
Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[12] |
Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206 |
[13] |
T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054 |
[14] |
Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001 |
[15] |
Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269 |
[16] |
T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067 |
[17] |
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 |
[18] |
Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077 |
[19] |
Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013 |
[20] |
Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]