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January  2019, 39(1): 395-430. doi: 10.3934/dcds.2019016

On the convergence of a stochastic 3D globally modified two-phase flow model

Theodore Tachim Medjo

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

Received  March 2018 Revised  July 2018 Published  October 2018

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.

Keywords: Allen-Cahn-Navier-Stokes, globally modified, stochastic, Galerkin scheme.
Mathematics Subject Classification: 35R60, 35Q35, 60H15, 76M35, 86A05.
Citation: Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016
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. 

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