doi: 10.3934/dcdsb.2021282
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On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects

Department of Mathematics, Florida International University, DM413B, MMC, Miami, Florida 33199, USA

Received  April 2021 Revised  September 2021 Early access November 2021

We consider a stochastic Allen-Cahn-Navier-Stokes equations with inertial effects in a bounded domain $ D\subset\mathbb{R}^{d} $, $ d = 2, 3 $, driven by a multiplicative noise. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients. The construction of the solution is based on the Faedo-Galerkin approximation, compactness method and the Skorokhod representation theorem.

Citation: T. Tachim Medjo. On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021282
References:
[1]

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.

[2]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[3]

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

[4]

A. Bensoussan and R. Temam, Équations stochastiques de type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.

[5]

G. BonfantiM. Frémond and F. Luterotti, Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear Anal. Real World Appl., 5 (2004), 123-140.  doi: 10.1016/S1468-1218(03)00021-X.

[6]

Z. BrzeźniakB. Goldys and T. Jegaraj, Weak solution of a stochastic landau-Lifshitz-Gilbert equation, Appl. Math. Res., 2013 (2013), 1-33.  doi: 10.1093/amrx/abs009.

[7]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput., 7 (2019), 417-475.  doi: 10.1007/s40072-018-0131-z.

[8]

E. Feireisl D. Breit and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter Series in Applied and Numerical Mathematics, 3. De Gruyter, Berlin, 2018.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.

[10]

G. Deugoué and J. K. Djoko, On the time discretization for the globally modified 3-dimensional Navier-Stokes equations, J. Comput. Appl. Math., 235 (2011), 2015-2029.  doi: 10.1016/j.cam.2010.10.003.

[11]

G. DeugouéB. Jidjou Moghomye and T. Tachim Medjo, Existence of a solution to the stochastic nonlocal Cahn-Hilliard Navier-Stokes model via a splitting-up method, Nonlinearity, 33 (2020), 3424-3469.  doi: 10.1088/1361-6544/ab8020.

[12]

G. DeugouéA. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise of jump type, Phys. D, 398 (2019), 23-68.  doi: 10.1016/j.physd.2019.05.012.

[13]

G. DeugouéA. Ndongmo Ngana and T. Tachim Medjo, On the strong solutions for a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes model, Dyn. Partial Differ. Equ., 17 (2020), 19-60.  doi: 10.4310/DPDE.2020.v17.n1.a2.

[14]

G. DeugouéP. A. Razafimandimby and M. Sango, On the 3-D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Process. Appl., 122 (2012), 2211-2248.  doi: 10.1016/j.spa.2012.03.002.

[15]

G. Deugoué and T. Tachim Medjo, On the convergence for the 3D globally modified Cahn-Hilliard-Navier-Stokes equations, J. Differential Equations, 265 (2018), 545-592. 

[16]

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.

[17]

G. Deugoué and T. Tachim Medjo, On a stochastic 2D Cahn-Hilliard-Navier-Stokes system driven by jump noise, Commun. Stoch. Anal., 13 (2019), Art. 5, 29 pp. doi: 10.31390/cosa.13.1.05.

[18]

G. Favre and G. Schimperna, On a Navier-Stokes-Allen-Cahn model with inertial effects, J. Math. Anal. Appl., 475 (2019), 811-838.  doi: 10.1016/j.jmaa.2019.02.074.

[19]

E. FeireislE. RoccaG. Schimperna and A. Zarnescu, On a hyperbolic system arising in liquid crystals modeling, J. Hyperbolic Differ. Equ., 15 (2018), 15-35.  doi: 10.1142/S0219891618500029.

[20]

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.

[21]

M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002.

[22]

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.

[23]

C. 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.

[24]

C. 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.

[25]

F. Gay-Balmaz and C. Tronci, The helicity and vorticity of liquid-crystal flows, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1197-1213.  doi: 10.1098/rspa.2010.0309.

[26]

M. GrasselliA. MiranvilleV. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509.  doi: 10.1002/mana.200510560.

[27]

M. Grasselli and V. Pata, Existence of a universal attractor for a parabolic-hyperbolic phase-field system, Adv. Math. Sci. Appl., 13 (2003), 443-459. 

[28]

M. GrasselliG. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170.  doi: 10.1080/03605300802608247.

[29]

M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys., 62 (2011), 979-992.  doi: 10.1007/s00033-011-0157-9.

[30]

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

[31]

O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (New York), Springer-Verlag, New York, 1997.

[32]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1970.

[33]

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

[34]

E. Motyl, Stochastic Navier-Stokes equations driven by L$\acute{e}$vy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.

[35]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London 1967.

[36]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007.

[37]

P. A. Razafimandimby and M. Sango, Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations, Z. Angew. Math. Phys., 66 (2015), 2197-2235.  doi: 10.1007/s00033-015-0534-x.

[38]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Phys. D, 239 (2010), 912-923.  doi: 10.1016/j.physd.2010.01.009.

[39]

A. V. Skorokhod, Studies in the Theory of Random Processes. Translated from the Russian by Scripta-Technica, Inc., 2$^{nd}$ edition, Addison-Wesley Publishing Co., Reading, Mass, 1965.

[40]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model, J. Differential Equations, 263 (2017), 1028-1054.  doi: 10.1016/j.jde.2017.03.008.

[41]

T. Tachim Medjo, A note on the regularity of weak solutions to the coupled 2D {A}llen-Cahn-Navier-Stokes system, J. Appl. Anal., 15 (2019), 111-117.  doi: 10.1515/jaa-2019-0012.

[42]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Allen-Cahn-Navier-Stokes model, Stoch. Dyn., 19 (2019), 1950007, 28pp. doi: 10.1142/S0219493719500072.

[43]

T. Tachim Medjo, On the weak solutions to a 3D stochastic Cahn-Hilliard-Navier-Stokes model, Z. Angew. Math. Phys., 71 (2020), 23pp. doi: 10.1007/s00033-019-1237-5.

[44]

R. Temam, Infinite Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997.

[45]

A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics, Springer-Verlag, New York, 1996.

show all references

References:
[1]

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.

[2]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[3]

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

[4]

A. Bensoussan and R. Temam, Équations stochastiques de type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.

[5]

G. BonfantiM. Frémond and F. Luterotti, Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear Anal. Real World Appl., 5 (2004), 123-140.  doi: 10.1016/S1468-1218(03)00021-X.

[6]

Z. BrzeźniakB. Goldys and T. Jegaraj, Weak solution of a stochastic landau-Lifshitz-Gilbert equation, Appl. Math. Res., 2013 (2013), 1-33.  doi: 10.1093/amrx/abs009.

[7]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput., 7 (2019), 417-475.  doi: 10.1007/s40072-018-0131-z.

[8]

E. Feireisl D. Breit and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter Series in Applied and Numerical Mathematics, 3. De Gruyter, Berlin, 2018.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.

[10]

G. Deugoué and J. K. Djoko, On the time discretization for the globally modified 3-dimensional Navier-Stokes equations, J. Comput. Appl. Math., 235 (2011), 2015-2029.  doi: 10.1016/j.cam.2010.10.003.

[11]

G. DeugouéB. Jidjou Moghomye and T. Tachim Medjo, Existence of a solution to the stochastic nonlocal Cahn-Hilliard Navier-Stokes model via a splitting-up method, Nonlinearity, 33 (2020), 3424-3469.  doi: 10.1088/1361-6544/ab8020.

[12]

G. DeugouéA. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise of jump type, Phys. D, 398 (2019), 23-68.  doi: 10.1016/j.physd.2019.05.012.

[13]

G. DeugouéA. Ndongmo Ngana and T. Tachim Medjo, On the strong solutions for a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes model, Dyn. Partial Differ. Equ., 17 (2020), 19-60.  doi: 10.4310/DPDE.2020.v17.n1.a2.

[14]

G. DeugouéP. A. Razafimandimby and M. Sango, On the 3-D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Process. Appl., 122 (2012), 2211-2248.  doi: 10.1016/j.spa.2012.03.002.

[15]

G. Deugoué and T. Tachim Medjo, On the convergence for the 3D globally modified Cahn-Hilliard-Navier-Stokes equations, J. Differential Equations, 265 (2018), 545-592. 

[16]

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.

[17]

G. Deugoué and T. Tachim Medjo, On a stochastic 2D Cahn-Hilliard-Navier-Stokes system driven by jump noise, Commun. Stoch. Anal., 13 (2019), Art. 5, 29 pp. doi: 10.31390/cosa.13.1.05.

[18]

G. Favre and G. Schimperna, On a Navier-Stokes-Allen-Cahn model with inertial effects, J. Math. Anal. Appl., 475 (2019), 811-838.  doi: 10.1016/j.jmaa.2019.02.074.

[19]

E. FeireislE. RoccaG. Schimperna and A. Zarnescu, On a hyperbolic system arising in liquid crystals modeling, J. Hyperbolic Differ. Equ., 15 (2018), 15-35.  doi: 10.1142/S0219891618500029.

[20]

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.

[21]

M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002.

[22]

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.

[23]

C. 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.

[24]

C. 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.

[25]

F. Gay-Balmaz and C. Tronci, The helicity and vorticity of liquid-crystal flows, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1197-1213.  doi: 10.1098/rspa.2010.0309.

[26]

M. GrasselliA. MiranvilleV. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509.  doi: 10.1002/mana.200510560.

[27]

M. Grasselli and V. Pata, Existence of a universal attractor for a parabolic-hyperbolic phase-field system, Adv. Math. Sci. Appl., 13 (2003), 443-459. 

[28]

M. GrasselliG. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170.  doi: 10.1080/03605300802608247.

[29]

M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys., 62 (2011), 979-992.  doi: 10.1007/s00033-011-0157-9.

[30]

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

[31]

O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (New York), Springer-Verlag, New York, 1997.

[32]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1970.

[33]

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

[34]

E. Motyl, Stochastic Navier-Stokes equations driven by L$\acute{e}$vy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.

[35]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London 1967.

[36]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007.

[37]

P. A. Razafimandimby and M. Sango, Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations, Z. Angew. Math. Phys., 66 (2015), 2197-2235.  doi: 10.1007/s00033-015-0534-x.

[38]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Phys. D, 239 (2010), 912-923.  doi: 10.1016/j.physd.2010.01.009.

[39]

A. V. Skorokhod, Studies in the Theory of Random Processes. Translated from the Russian by Scripta-Technica, Inc., 2$^{nd}$ edition, Addison-Wesley Publishing Co., Reading, Mass, 1965.

[40]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model, J. Differential Equations, 263 (2017), 1028-1054.  doi: 10.1016/j.jde.2017.03.008.

[41]

T. Tachim Medjo, A note on the regularity of weak solutions to the coupled 2D {A}llen-Cahn-Navier-Stokes system, J. Appl. Anal., 15 (2019), 111-117.  doi: 10.1515/jaa-2019-0012.

[42]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Allen-Cahn-Navier-Stokes model, Stoch. Dyn., 19 (2019), 1950007, 28pp. doi: 10.1142/S0219493719500072.

[43]

T. Tachim Medjo, On the weak solutions to a 3D stochastic Cahn-Hilliard-Navier-Stokes model, Z. Angew. Math. Phys., 71 (2020), 23pp. doi: 10.1007/s00033-019-1237-5.

[44]

R. Temam, Infinite Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997.

[45]

A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics, Springer-Verlag, New York, 1996.

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