We consider the stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations on a bounded domain , , 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.
AbelsH. (2009a). On a diffusive interface model for two-phase flows of viscous, incompressible fluids with mathched densities. Archive for Rational Mechanics and Analysis, 194, 463–506.
2.
AbelsH. (2009b). Longtime behavior of solutions of a Navier–Stokes/Cahn–Hilliard system. Banach Center Publications, 86, 9–19. Proceedings of the Conference “Nonlocal and Abstract Parabolic Equations and their Applications,” Bedlewo.
3.
BatesP. W.HanJ. (2005). The Dirichlet boundary problem for a nonlocal Cahn–Hilliard equations. Journal of Mathematical Analysis and Applications, 331, 289–312.
BensoussanA.TemamR. (1973). Equations stochastiques du type Navier–Stokes. Journal of Functional Analysis, 13, 195–222.
6.
BrzeźniakZ.MotylE. (2013). Existence of a Martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains. The Journal of Differential Equations, 254, 1627–1685.
7.
CaoC.GalC. G. (2012). Global solutions for the 2D NS–CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility. Nonlinearity, 25, 3211–3234.
8.
ColliP.FrigeriS.GrasselliM. (2012). Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system. Journal of Mathematical Analysis and Applications, 386, 428–444.
9.
Da PratoG.ZabczykJ. (2014). Stochastic equations in infinite dimensions, 2 editions, encyclopedia of mathematics and its applications (Vol. 152). Cambridge University Press.
10.
DeugouéG.SangoM. (2010). On the strong solution for the 3D Stochastic Leray-Alpha model. Boundary Value Problems, Article ID 723018, (p. 31).
11.
DeugouéG.Tachim MedjoT. (2018a). The exponential behavior of a stochastic globally modified Cahn–Hilliard–Navier–Stokes model with multiplicative noise. Journal of Mathematical Analysis and Applications, 460(1), 140–163.
12.
DeugouéG.Tachim MedjoT. (2018b). Convergence of the solution of the stochastic 3D globally modified Cahn–Hilliard–Navier–Stokes equations. The Journal of Differential Equations, 265(2), 545–592.
13.
FlandoliF.GatarekD. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probability Theory and Related Fields, 102(3), 367–391.
14.
FrigeriS. (2016). Global existence of weak solutions for a nonlocal model for two-phase flows of imcompressible fluids with unmatched densities. Mathematical Models & Methods in Applied Sciences, 26, 1955–1993.
15.
FrigeriS.GalC. G.GrasselliM. (2016). On nonlocal Cahn–Hilliard–Navier–Stokes systems in two dimensions. Journal of Nonlinear Science, 26(4), 847–893.
16.
FrigeriS.GrasselliM. (2002). Global and trajectories attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system. Journal of Dynamics and Differential Equations, 24, 827–856.
17.
FrigeriS.GrasselliM.KrëjěiP. (2013). Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems. The Journal of Differential Equations, 255(9), 2597–2614.
18.
GajewskiH.ZachariasK. (2003). On a nonlocal phase separation model. Journal of Mathematical Analysis and Applications, 286, 11–31.
19.
GalC. G.GrasselliM. (2010a). Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 27, 401–436.
20.
GalC. G.GrasselliM. (2010b). Trajectory attractors for binary fluid mixture in 3D. Chinese Annals of Mathematics, Series B, 31, 655–678.
21.
GalC. G.GrasselliM. (2011). Instability of two phase flows: A lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system. Physica D, 240, 629–635.
22.
GalC. G.GrasselliM. (2014). Longtime behavior of nonlocal Cahn–Hilliard equations. Discrete and Continuous Dynamical Systems Series A, 34, 145–179.
23.
GiacominG.LebowitzJ. L. (1996). Exact macroscopic description of phase segregation in model alloys with long range interactions. Physical Review Letters, 76, 1094–1097.
24.
GiacominG.LebowitzJ. L. (1997). Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic Limits. Journal of Statistical Physics, 87, 37–61.
25.
GiacominG.LebowitzJ. L. (1998). Phase segregation dynamics in particle systems with long range interactions. II. Phase Motion. SIAM Journal of Applied Mathematics, 58, 1707–1729.
26.
HaroskeD. D.TriebelH. (2008). Distributions, Sobolev spaces, Elliptic Equations. European Mathematical Society. EMS Textbooks in Mathematics.
27.
HollyK.WiciakM. (1995). Compactness method applied to an abstract nonlinear parabolic equation. In Selected problems of mathematics (pp. 95–160). Cracow University ot Technology.
28.
IkedaN.WatanabeS. (1989). Stochastic differential equations and diffusion processes. North-Holland Publishing Co. Second edition. vol. 24 of North-Holland Mathematical Library.
29.
KallenbergO. (1997). Foundations of modern probability, in, probability and its applications. Springer-Verlag.
30.
KimJ. S. (2012). Phase-field models for multi-component fluid flows. Communications in Computational Physics, 12, 613–661.
31.
LiF.YouB. (2018). Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise. Stochastic Analysis and Applications, 36(3), 546–559.
32.
LionsJ. L. (1969). Quelques méthodes de resolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars.
33.
OuhabazE. M. (2005). Analysis of heat equations on domains. Princeton University Press. 31 of London Mathematical society monographs Series.
34.
PrévôtC.RöcknerM. (2007). A concise course on stochastic partial differential equations. Springer-Verlag. 1905 of Lecture Notes in Mathematics.
35.
ProhorovYu. V. (1956). Convergence of random processes and limits theorems in probability theory. (Russian), Teorija Verojatnosteii Primenenija, 1, 177–238.
36.
SkorokhodA. V. (1956). Limits theorem for stochastic processes. Teorija Verojatnosteii Primenenija, 1, 289–319.
37.
SkorokhodA. V. (1965). Addison-Wesley Publishing Co. Translated from the Russian by Scripta-Technica, Inc.
38.
StarovoitovV. N. (1997). The dynamics of a two-component fluid in the presence of capillary forces. Mathematical Notes, 62, 244–254.
39.
Tachim MedjoT. (2017a). On the existence and uniqueness of solution to a stochastic 2D Cahn–Hilliard–Navier–Stokes model. The Journal of Differential Equations, 262, 1028–1054.
40.
Tachim MedjoT. (2017b). Unique strong and -attractor of a three dimensional globall, modified Cahn–Hilliard–Navier–Stokes model. Applicable Analysis, 96(16), 2695–2716.
41.
Tachim MedjoT. (2018). Pullback V-attractor for a three dimensional globally modified Cahn–Hilliard–Navier–Stokes model. Applicable Analysis, 97(6), 1016–1041.
42.
TemamR. (1984). Navier–Stokes equations: Theory and numerical analysis (3rd ed.). 2 of Studies in Mathematics and Its Applications. AMS Chelsea Publishing.
43.
TemamR. (1995). Navier–Stokes equations and nonlinear functional analysis, second edition, CBMS-NSF regional conference series in applied mathematics. SIAM.
44.
ZhaoL.WuH.HuangH. (2009). Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids. Communications in Mathematical Sciences, 7, 939–962.
