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May 2019, 18(3): 1117-1138. doi: 10.3934/cpaa.2019054

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

Department of Mathematics and Satistics, Florida International University, MMC, Miami, Florida 33199, USA

Received  January 2018 Revised  June 2018 Published  November 2018

In this article, we study the stability of weak solutions to a stochastic version of a coupled Cahn-Hilliard-Navier-Stokes model with multiplicative noise. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.

Citation: T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054
References:
[1]

H. Abels, 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.

[2]

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.

[3]

C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension, Adv. Math. Sci. Appl., 4 (1984), 465-489.

[4]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123.

[5]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Anal., 20 (1999), 175-212.

[6]

F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9.

[7]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Computer and Fluids, 31 (2002), 41-68.

[8]

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.

[9]

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.

[10]

C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233. doi: 10.1002/cpa.10056.

[11]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[12]

T. CaraballoJ. Langa and T. Taniguchi, The exponential behavior and stabilizability of stochastic 2D-Navier-Stokes equations, J. Differential Equations, 179 (2002), 714-737. doi: 10.1006/jdeq.2001.4037.

[13]

T. CaraballoA. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D LANS-α model, Appl. Math. Optim., 53 (2006), 141-161. doi: 10.1007/s00245-005-0839-9.

[14]

T. CaraballoJ. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 459-479. doi: 10.1098/rspa.2005.1574.

[15] T. DuboisF. Jauberteau and R. Temam, Dynamic Multilevel Methods and the Numerical Simulation of Turbulence, Cambridge University Press, Cambridge, 1999.
[16]

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.

[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), 655-678. doi: 10.1007/s11401-010-0603-6.

[19]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159.

[20]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. doi: 10.1088/0951-7715/20/12/001.

[21]

B. Di Martino and P. Orenga, Resolution to a three-dimensional physical oceanographic problem using the non-linear Galerkin method, Int. J. Num. Meth. Fluids, 30 (1999), 577-606. doi: 10.1002/(SICI)1097-0363(19990715)30:5<577::AID-FLD858>3.0.CO;2-T.

[22]

T. Tachim Medjo, On the weak solutions to a stochastic Cahn-Hilliard-Navier-Stokes model, Submitted.

[23]

T. Tachim Medjo, A small eddy correction method for a 3D Navier-Stokes type equations related to the primitive equations of the ocean, SIAM J. Numer. Anal., 45 (2007), 1843-1870. doi: 10.1137/05063074X.

[24]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model, J. Differential Equations, 262 (2017), 1028-1054. doi: 10.1016/j.jde.2017.03.008.

[25]

G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains i: Global attractors and global regularity of solutions, J. Amer. Math. Society, 6 (1993), 503-568. doi: 10.2307/2152776.

[26]

P. A. Razafimandimby and M. Sango, On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids, Appl. Anal., 91 (2012), 2217-2233. doi: 10.1080/00036811.2011.598861.

[27]

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.

[28]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001. doi: 10.1090/chel/343.

show all references

References:
[1]

H. Abels, 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.

[2]

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.

[3]

C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension, Adv. Math. Sci. Appl., 4 (1984), 465-489.

[4]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123.

[5]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Anal., 20 (1999), 175-212.

[6]

F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9.

[7]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Computer and Fluids, 31 (2002), 41-68.

[8]

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.

[9]

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.

[10]

C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233. doi: 10.1002/cpa.10056.

[11]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[12]

T. CaraballoJ. Langa and T. Taniguchi, The exponential behavior and stabilizability of stochastic 2D-Navier-Stokes equations, J. Differential Equations, 179 (2002), 714-737. doi: 10.1006/jdeq.2001.4037.

[13]

T. CaraballoA. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D LANS-α model, Appl. Math. Optim., 53 (2006), 141-161. doi: 10.1007/s00245-005-0839-9.

[14]

T. CaraballoJ. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 459-479. doi: 10.1098/rspa.2005.1574.

[15] T. DuboisF. Jauberteau and R. Temam, Dynamic Multilevel Methods and the Numerical Simulation of Turbulence, Cambridge University Press, Cambridge, 1999.
[16]

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.

[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), 655-678. doi: 10.1007/s11401-010-0603-6.

[19]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159.

[20]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. doi: 10.1088/0951-7715/20/12/001.

[21]

B. Di Martino and P. Orenga, Resolution to a three-dimensional physical oceanographic problem using the non-linear Galerkin method, Int. J. Num. Meth. Fluids, 30 (1999), 577-606. doi: 10.1002/(SICI)1097-0363(19990715)30:5<577::AID-FLD858>3.0.CO;2-T.

[22]

T. Tachim Medjo, On the weak solutions to a stochastic Cahn-Hilliard-Navier-Stokes model, Submitted.

[23]

T. Tachim Medjo, A small eddy correction method for a 3D Navier-Stokes type equations related to the primitive equations of the ocean, SIAM J. Numer. Anal., 45 (2007), 1843-1870. doi: 10.1137/05063074X.

[24]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model, J. Differential Equations, 262 (2017), 1028-1054. doi: 10.1016/j.jde.2017.03.008.

[25]

G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains i: Global attractors and global regularity of solutions, J. Amer. Math. Society, 6 (1993), 503-568. doi: 10.2307/2152776.

[26]

P. A. Razafimandimby and M. Sango, On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids, Appl. Anal., 91 (2012), 2217-2233. doi: 10.1080/00036811.2011.598861.

[27]

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.

[28]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001. doi: 10.1090/chel/343.

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