# American Institute of Mathematical Sciences

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

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

 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  March 2021 Early access  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.

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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