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March  2021, 20(3): 1135-1170. doi: 10.3934/cpaa.2021010

Approximation of a stochastic two-phase flow model by a splitting-up method

G. Deugoué 1, , B. Jidjou Moghomye 1, and  T. Tachim Medjo 2,,

1. 

Department of Mathematics and Computer Science, University of Dschang, P. O. BOX 67, Dschang, Cameroon
2. 

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

* Corresponding author

Received  September 2019 Revised  October 2020 Published  February 2021

Fund Project: The first author is supported by the Fulbright Scholar Program Advanced Research and the Florida International University, 2019

In this paper, we consider a stochastic Allen-Cahn Navier-Stokes system in a bounded domain of $ \mathbb{R}^d, $ $ d = 2,3 $. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on splitting-up method as well as some compactness method.

Keywords: Stochastic Navier-Stokes, Allen-Cahn, weak martingale solutions, splitting-up method, cylindrical Wiener process, compactness.
Mathematics Subject Classification: 35R60, 35Q35, 60H15, 76M35, 86A05.
Citation: G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010
References:
[1]

D. C. Antonopoulou, G. Karali and A. Millet, Existence and regularity of solution for a stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion, J. Differ. Equ., 260 (2016), 2383-2417.  doi: 10.1016/j.jde.2015.10.004.  Google Scholar

[2]

A. Bensoussan, Some existence results for stochastic partial differential equations. In Stochastic Partial Differential Equations and Applications, Pitman Res.Notes, Math. Ser., 268, Longman Scientific and Technical, Harlow, UK, (1992), 37-53.  Google Scholar

[3]

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

[4]

A. Bensoussan, R. Glowinski and A. Rascanu, Approximation of some stochastic differential equations by the splitting-up method, Appl. Math. Optim., 25 (1992), 81-106.  doi: 10.1007/BF01184157.  Google Scholar

[5]

P. Billingsley, Convergence of Probability Measures, 2nd edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[6]

Z. Brzeźniak, B. Goldys and T. Jegaraj, Weak solutions of a stochastic Landau-Lifshitz-Gilbert equation, Appl. Math. Res. Express., 1 (2013), 1-33.  doi: 10.1093/amrx/abs009.  Google Scholar

[7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions: Second Edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[8]

S. Dai and Q. Du, Weak Solutions for the Cahn-Hilliard Equation with Degenerate Mobility, Arch. Rational Mech. Anal., 219 (2016), 1161-1184.  doi: 10.1007/s00205-015-0918-2.  Google Scholar

[9]

G. Deugoue and M. Sango, Convergence for a Splitting-Up Scheme for the 3D Stochastic Navier-Stokes-$\alpha$ Model, Stoch. Anal. Appl., 32 (2014), 253-279.  doi: 10.1080/07362994.2013.862359.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical society, 1997.  Google Scholar

[11]

X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase, Fluids. Math. Comput., 76 (2007), 539-571.  doi: 10.1090/S0025-5718-06-01915-6.  Google Scholar

[12]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[13]

G. B. Folland, Real analysis. Pure and Applied Mathematics, John Wiley and Sons Inc, New York, 1999.  Google Scholar

[14]

C. G. Gal and M. Grasselli, behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. S., 28 (2010), 1-39.  doi: 10.3934/dcds.2010.28.1.  Google Scholar

[15]

N. Y. Goncharuk and P. Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Processes Appl., 73 (1998), 1-45.  doi: 10.1016/S0304-4149(97)00079-3.  Google Scholar

[16]

L. Goudenège and L. Manca, Stochastic phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model, arXiv: 1901.01335. Google Scholar

[17]

W. Grecksch, A splitting up method for nonlinear parabolic Ito equations, preprint, Martin-Luther-Universitat, Halle-Wittenberg, 1996. Google Scholar

[18]

I. Gyöngy and N. Krylov, On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591.  doi: 10.1214/aop/1048516528.  Google Scholar

[19]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximations by Fourier-spectral method, Phys. D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.  Google Scholar

[20]

N. Nagase, Remarks on nonlinear stochastic partial differential equations: an application of the splitting-up method, SIAM J. Control Optim., 33 (1995), 1716-1730.  doi: 10.1137/S036301299324618X.  Google Scholar

[21]

E. Pardoux, Equations and Dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse Université Paris XI, 1975. Google Scholar

[22] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, Academic Press, Inc., New York-London, 1967.   Google Scholar
[23]

D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edition, in Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[24]

M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931-965.  doi: 10.1515/form.2011.138.  Google Scholar

[25]

T. Tachim Medjo, On the convergence of a stochastic 3D globally modified two-phase flow model, Discret. Contin. Dyn. S., 39 (2019), 395-430.  doi: 10.3934/dcds.2019016.  Google Scholar

[26]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Allen-Cahn-Navier-Stokes model, Stoch. Dynam., 19 (2018), 28 pp. doi: 10.1142/S0219493719500072.  Google Scholar

[27]

T. Tachim Medjo, A two-phase flow model with delays, Discrete Cont. Dyn-B, 22 (2017), 1-17.  doi: 10.3934/dcdsb.2017137.  Google Scholar

[28]

T. Tachim Medjo, Pullback $\mathbb{V}$-attractor of a three dimensional globally modified two-phase flow model, Discrete Cont. Dyn. S., 38 (2018), 2141-2169.  doi: 10.3934/dcds.2018088.  Google Scholar

[29]

T. Tachim Medjo, C. Tone and F. Tone, Long-time dynamics of a regularized family of models for homogeneous incompressible two-phase flows, Asymptotic Anal., 94 (2015), 125-160.  doi: 10.3233/ASY-151309.  Google Scholar

show all references

References:
[1]

D. C. Antonopoulou, G. Karali and A. Millet, Existence and regularity of solution for a stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion, J. Differ. Equ., 260 (2016), 2383-2417.  doi: 10.1016/j.jde.2015.10.004.  Google Scholar

[2]

A. Bensoussan, Some existence results for stochastic partial differential equations. In Stochastic Partial Differential Equations and Applications, Pitman Res.Notes, Math. Ser., 268, Longman Scientific and Technical, Harlow, UK, (1992), 37-53.  Google Scholar

[3]

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

[4]

A. Bensoussan, R. Glowinski and A. Rascanu, Approximation of some stochastic differential equations by the splitting-up method, Appl. Math. Optim., 25 (1992), 81-106.  doi: 10.1007/BF01184157.  Google Scholar

[5]

P. Billingsley, Convergence of Probability Measures, 2nd edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[6]

Z. Brzeźniak, B. Goldys and T. Jegaraj, Weak solutions of a stochastic Landau-Lifshitz-Gilbert equation, Appl. Math. Res. Express., 1 (2013), 1-33.  doi: 10.1093/amrx/abs009.  Google Scholar

[7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions: Second Edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[8]

S. Dai and Q. Du, Weak Solutions for the Cahn-Hilliard Equation with Degenerate Mobility, Arch. Rational Mech. Anal., 219 (2016), 1161-1184.  doi: 10.1007/s00205-015-0918-2.  Google Scholar

[9]

G. Deugoue and M. Sango, Convergence for a Splitting-Up Scheme for the 3D Stochastic Navier-Stokes-$\alpha$ Model, Stoch. Anal. Appl., 32 (2014), 253-279.  doi: 10.1080/07362994.2013.862359.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical society, 1997.  Google Scholar

[11]

X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase, Fluids. Math. Comput., 76 (2007), 539-571.  doi: 10.1090/S0025-5718-06-01915-6.  Google Scholar

[12]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[13]

G. B. Folland, Real analysis. Pure and Applied Mathematics, John Wiley and Sons Inc, New York, 1999.  Google Scholar

[14]

C. G. Gal and M. Grasselli, behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. S., 28 (2010), 1-39.  doi: 10.3934/dcds.2010.28.1.  Google Scholar

[15]

N. Y. Goncharuk and P. Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Processes Appl., 73 (1998), 1-45.  doi: 10.1016/S0304-4149(97)00079-3.  Google Scholar

[16]

L. Goudenège and L. Manca, Stochastic phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model, arXiv: 1901.01335. Google Scholar

[17]

W. Grecksch, A splitting up method for nonlinear parabolic Ito equations, preprint, Martin-Luther-Universitat, Halle-Wittenberg, 1996. Google Scholar

[18]

I. Gyöngy and N. Krylov, On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591.  doi: 10.1214/aop/1048516528.  Google Scholar

[19]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximations by Fourier-spectral method, Phys. D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.  Google Scholar

[20]

N. Nagase, Remarks on nonlinear stochastic partial differential equations: an application of the splitting-up method, SIAM J. Control Optim., 33 (1995), 1716-1730.  doi: 10.1137/S036301299324618X.  Google Scholar

[21]

E. Pardoux, Equations and Dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse Université Paris XI, 1975. Google Scholar

[22] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, Academic Press, Inc., New York-London, 1967.   Google Scholar
[23]

D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edition, in Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[24]

M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931-965.  doi: 10.1515/form.2011.138.  Google Scholar

[25]

T. Tachim Medjo, On the convergence of a stochastic 3D globally modified two-phase flow model, Discret. Contin. Dyn. S., 39 (2019), 395-430.  doi: 10.3934/dcds.2019016.  Google Scholar

[26]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Allen-Cahn-Navier-Stokes model, Stoch. Dynam., 19 (2018), 28 pp. doi: 10.1142/S0219493719500072.  Google Scholar

[27]

T. Tachim Medjo, A two-phase flow model with delays, Discrete Cont. Dyn-B, 22 (2017), 1-17.  doi: 10.3934/dcdsb.2017137.  Google Scholar

[28]

T. Tachim Medjo, Pullback $\mathbb{V}$-attractor of a three dimensional globally modified two-phase flow model, Discrete Cont. Dyn. S., 38 (2018), 2141-2169.  doi: 10.3934/dcds.2018088.  Google Scholar

[29]

T. Tachim Medjo, C. Tone and F. Tone, Long-time dynamics of a regularized family of models for homogeneous incompressible two-phase flows, Asymptotic Anal., 94 (2015), 125-160.  doi: 10.3233/ASY-151309.  Google Scholar

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