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

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

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.

Citation: Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

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

[3]

D. BreitE. 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. Google Scholar

[4]

Z. BrzeźiakW 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. Google Scholar

[5]

Z. BrzeźniakE. 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. Google Scholar

[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. Google Scholar

[7]

T. CaraballoJ. 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. Google Scholar

[8]

S. ChenC. FoiasD. D. HolmE. OlsonE. 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. Google Scholar

[9]

S. ChenC. FoiasD. D. HolmE. OlsonE. 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. Google Scholar

[10]

S. ChenC. FoiasD. D. HolmE. OlsonE. 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. Google Scholar

[11]

S. ChenD. D. HolmL. 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. Google Scholar

[12]

A. DebusscheN. 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. Google Scholar

[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. Google Scholar

[14]

E. FeireislH. PetzeltováE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

N. Glatz-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600. Google Scholar

[21]

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

[22]

D. D. HolmJ. 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. Google Scholar

[23]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. Google Scholar

[24]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland, Kodansha, 1989. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781107295513. Google Scholar

[29]

C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, 2007. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

A. V. Skorohod I. I. Gikhman, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

M. I. VishikA. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210,256. Google Scholar

show all references

References:
[1]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304. doi: 10.1007/BF00996149. Google Scholar

[2]

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

[3]

D. BreitE. 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. Google Scholar

[4]

Z. BrzeźiakW 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. Google Scholar

[5]

Z. BrzeźniakE. 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. Google Scholar

[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. Google Scholar

[7]

T. CaraballoJ. 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. Google Scholar

[8]

S. ChenC. FoiasD. D. HolmE. OlsonE. 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. Google Scholar

[9]

S. ChenC. FoiasD. D. HolmE. OlsonE. 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. Google Scholar

[10]

S. ChenC. FoiasD. D. HolmE. OlsonE. 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. Google Scholar

[11]

S. ChenD. D. HolmL. 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. Google Scholar

[12]

A. DebusscheN. 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. Google Scholar

[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. Google Scholar

[14]

E. FeireislH. PetzeltováE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

N. Glatz-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600. Google Scholar

[21]

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

[22]

D. D. HolmJ. 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. Google Scholar

[23]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. Google Scholar

[24]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland, Kodansha, 1989. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781107295513. Google Scholar

[29]

C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, 2007. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

A. V. Skorohod I. I. Gikhman, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

M. I. VishikA. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210,256. Google Scholar

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