A stochastic Allen-Cahn-Navier-Stokes system with singular potential. (English) Zbl 1533.35266
Summary: We investigate a stochastic version of the Allen-Cahn-Navier-Stokes system in a smooth two- or three-dimensional domain with random initial data. The system consists of a Navier-Stokes equation coupled with a convective Allen-Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. The problem is endowed with a no-slip boundary condition for the (volume averaged) velocity field, as well as a homogeneous Neumann condition for the order parameter. We first prove the existence of analytically weak martingale solutions in two and three spatial dimensions. Then, in two dimensions, we also establish pathwise uniqueness and the existence of a unique probabilistically-strong solution. Eventually, by exploiting a suitable generalisation of the classical De Rham theorem to stochastic processes, existence and uniqueness of a pressure is also shown.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76T06 | Liquid-liquid two component flows |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
Allen-Cahn-Navier-Stokes system; stochastic two-phase flow; logarithmic potential; martingale solutions; pathwise uniquenessReferences:
| [1] | Abels, H., On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal., 2, 463-506 (2009) · Zbl 1254.76158 |
| [2] | Abels, H.; Depner, D.; Garcke, H., Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech., 453-480 (2013) · Zbl 1273.76421 |
| [3] | Abels, H.; Feireisl, E., On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana Univ. Math. J., 2, 659-698 (2008) · Zbl 1144.35041 |
| [4] | Abels, H.; Garcke, H.; Grün, G., Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. (2012), 40 pp. · Zbl 1242.76342 |
| [5] | Allen, S. M.; Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall., 1085-1095 (1979) |
| [6] | Anderson, D.-M.; McFadden, G.-B.; Wheeler, A.-A., Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech., 139-165 (1997) · Zbl 1398.76051 |
| [7] | Antonopoulou, D. C.; Bates, P. W.; Blömker, D.; Karali, G. D., Motion of a droplet for the stochastic mass-conserving Allen-Cahn equation. SIAM J. Math. Anal., 1, 670-708 (2016) · Zbl 1439.35232 |
| [8] | Barbu, V., Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics (2010), Springer: Springer New York, pp. x+272 · Zbl 1197.35002 |
| [9] | Bauzet, C.; Bonetti, E.; Bonfanti, G.; Lebon, F.; Vallet, G., A global existence and uniqueness result for a stochastic Allen-Cahn equation with constraint. Math. Methods Appl. Sci., 14, 5241-5261 (2017) · Zbl 1390.35434 |
| [10] | Bertacco, F., Stochastic Allen-Cahn equation with logarithmic potential. Nonlinear Anal. (2021), 22 pp. · Zbl 1450.35306 |
| [11] | Bertacco, F.; Orrieri, C.; Scarpa, L., Random separation property for stochastic Allen-Cahn-type equations. Electron. J. Probab., 1-32 (2022) · Zbl 1498.35636 |
| [12] | Berti, A.; Berti, V.; Grandi, D., Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids. Nonlinearity, 11, 3143-3164 (2011) · Zbl 1269.35029 |
| [13] | Bertini, L.; Buttà, P.; Pisante, A., Stochastic Allen-Cahn equation with mobility. Nonlinear Differ. Equ. Appl., 54 (2017), 38 pp. · Zbl 1386.35497 |
| [14] | Blesgen, T., A generalization of the Navier-Stokes equations to two-phase flows. J. Phys. D, Appl. Phys., 10, 1119-1123 (1999) |
| [15] | Blömker, D.; Gawron, B.; Wanner, T., Nucleation in the one-dimensional stochastic Cahn-Hilliard model. Discrete Contin. Dyn. Syst., 1, 25-52 (2010) · Zbl 1196.35234 |
| [16] | Blömker, D.; Maier-Paape, S.; Wanner, T., Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation. Trans. Am. Math. Soc., 1, 449-489 (2008) · Zbl 1130.60066 |
| [17] | Brezis, H.; Mironescu, P., Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 5, 1355-1376 (2018) · Zbl 1401.46022 |
| [18] | Bréhier, C.-B.; Goudenège, L., Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete Contin. Dyn. Syst., Ser. B, 8, 4169-4190 (2019) · Zbl 07099047 |
| [19] | Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 258-267 (1958) · Zbl 1431.35066 |
| [20] | Cahn, J. W.; Hilliard, J. E., On spinodal decomposition. Acta Metall., 9, 795-801 (1961) |
| [21] | Cook, H. E., Brownian motion in spinodal decomposition. Acta Metall., 3, 297-306 (1970) |
| [22] | Da Prato, G.; Debussche, A., Stochastic Cahn-Hilliard equation. Nonlinear Anal., 2, 241-263 (1996) · Zbl 0838.60056 |
| [23] | Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications (2014), Cambridge University Press: Cambridge University Press Cambridge, pp. xviii+493 · Zbl 1317.60077 |
| [24] | Debussche, A.; Goudenège, L., Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections. SIAM J. Math. Anal., 3, 1473-1494 (2011) · Zbl 1233.60036 |
| [25] | Debussche, A.; Zambotti, L., Conservative stochastic Cahn-Hilliard equation with reflection. Ann. Probab., 5, 1706-1739 (2007) · Zbl 1130.60068 |
| [26] | Deugoué, G.; Moghomye, B. Jidjou; Tachim Medjo, T., Approximation of a stochastic two-phase flow model by a splitting-up method. Commun. Pure Appl. Anal., 1135-1170 (2021) · Zbl 1471.35231 |
| [27] | Deugoué, G.; Ngana, A. Ndongmo; Tachim Medjo, T., Strong solutions for the stochastic Cahn-Hilliard-Navier-Stokes system. J. Differ. Equ., 27-76 (2021) · Zbl 1455.35306 |
| [28] | Deugoué, G.; Tachim Medjo, T., Large deviation for a 2D Allen-Cahn-Navier-Stokes model under random influences. Asymptot. Anal., 1-2, 41-78 (2021) · Zbl 1473.35437 |
| [29] | Diestel, J.; Uhl, J., Vector Measures. Mathematical Surveys and Monographs (1977), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0369.46039 |
| [30] | Edwards, R. E., Functional Analysis. Theory and Applications (1965), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York-Toronto-London, xiii+781 · Zbl 0182.16101 |
| [31] | Elezović, N.; Mikelić, A., On the stochastic Cahn-Hilliard equation. Nonlinear Anal., 12, 1169-1200 (1991) · Zbl 0729.60057 |
| [32] | Feireisl, E.; Petcu, M., A diffuse interface model of a two-phase flow with thermal fluctuations. Appl. Math. Optim., 1, 531-563 (2021) · Zbl 1473.60097 |
| [33] | Feireisl, E.; Petcu, M., Stability of strong solutions for a model of incompressible two-phase flow under thermal fluctuations. J. Differ. Equ., 3, 1836-1858 (2019) · Zbl 1416.35204 |
| [34] | Feireisl, E.; Petcu, M.; Pražák, D., Relative energy approach to a diffuse interface model of a compressible two-phase flow. Math. Methods Appl. Sci., 5, 1465-1479 (2019) · Zbl 1420.35185 |
| [35] | Feireisl, E.; Rocca, E.; Petzeltová, E.; Schimperna, G., Analysis of a phase-field model for two-phase compressible fluids. Math. Models Methods Appl. Sci., 1129-1160 (2010) · Zbl 1200.76155 |
| [36] | Flandoli, F.; Gatarek, D., Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields, 3, 367-391 (1995) · Zbl 0831.60072 |
| [37] | Flory, P. J., Thermodynamics of high polymer solutions. J. Chem. Phys., 51-61 (1942) |
| [38] | Gal, C. G.; Grasselli, M., Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete Contin. Dyn. Syst., 1-39 (2010) · Zbl 1194.35056 |
| [39] | Gal, C. G.; Grasselli, M., Trajectory attractors for binary fluid mixtures in 3D. Chin. Ann. Math., Ser. B, 655-678 (2010) · Zbl 1223.35079 |
| [40] | Gal, C. G.; Grasselli, M.; Poiatti, A., Allen-Cahn-Navier-Stokes-Voigt systems with moving contact lines. J. Math. Fluid Mech. (2023), 57 pp. · Zbl 07766055 |
| [41] | Giga, M. H.; Kirshtein, A.; Liu, C., Variational Modeling and Complex Fluids. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2018), Springer: Springer Cham |
| [42] | Giorgini, A., Well-posedness of the two-dimensional Abels-Garcke-Grün model for two-phase flows with unmatched densities. Calc. Var. Partial Differ. Equ., 3 (2021), 40 pp. · Zbl 1468.35134 |
| [43] | Giorgini, A.; Grasselli, M.; Wu, H., Diffuse interface models for incompressible binary fluids and the mass-conserving Allen-Cahn approximation. J. Funct. Anal. (2022), 86 pp. |
| [44] | Giorgini, A.; Miranville, A.; Temam, R., Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system. SIAM J. Math. Anal., 2535-2574 (2019) · Zbl 1419.35160 |
| [45] | Giorgini, A.; Temam, R., Weak and strong solutions to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system. J. Math. Pures Appl. (9), 194-249 (2020) · Zbl 1452.35151 |
| [46] | Giorgini, A.; Temam, R.; Vu, X.-T., The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete Contin. Dyn. Syst., Ser. B, 1, 337-366 (2021) · Zbl 1464.35222 |
| [47] | Goudenège, L.; Manca, L., Stochastic phase field \(α\)-Navier-Stokes vesicle-fluid interaction model. J. Math. Anal. Appl. (2021), 39 pp. · Zbl 1462.35291 |
| [48] | Goudenège, L., Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection. Stoch. Process. Appl., 10, 3516-3548 (2009) · Zbl 1180.60054 |
| [49] | Gurtin, M. E.; Polignone, D.; Viñals, J., Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci., 6, 815-831 (1996) · Zbl 0857.76008 |
| [50] | Hairer, M.; Lê, K.; Rosati, T., The Allen-Cahn equation with generic initial datum. Probab. Theory Relat. Fields, 957-998 (2023) · Zbl 1521.60027 |
| [51] | Hairer, M.; Ryser, M.; Weber, H., Triviality of the 2D stochastic Allen-Cahn equation. Electron. J. Probab., 1-14 (2012) · Zbl 1245.60063 |
| [52] | Heida, M.; Málek, J.; Rajagopal, K. R., On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework. Z. Angew. Math. Phys., 759-776 (2012) · Zbl 1295.35360 |
| [53] | Heida, M.; Málek, J.; Rajagopal, K. R., On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework. Z. Angew. Math. Phys., 145-169 (2012) · Zbl 1295.76002 |
| [54] | Hohenberg, P. C.; Halperin, B. I., Theory of dynamic critical phenomena. Rev. Mod. Phys., 435-479 (1977) |
| [55] | Hošek, R.; Mácha, V., Weak-strong uniqueness for Allen-Cahn/Navier-Stokes system. Czechoslov. Math. J., 837-851 (2019) · Zbl 1513.35454 |
| [56] | Huggins, M. L., Solutions of long chain compounds. J. Chem. Phys., 440 (1941) |
| [57] | Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, pp. xvi+555 · Zbl 0205.43702 |
| [58] | Jiang, J.; Li, Y.-H.; Liu, C., Two-phase incompressible flows with variable density: an energetic variational approach. Discrete Contin. Dyn. Syst., 3243-3284 (2017) · Zbl 1361.35129 |
| [59] | Kotschote, M., Strong solutions of the Navier-Stokes equations for a compressible fluid of Allen-Cahn type. Arch. Ration. Mech. Anal., 489-514 (2012) · Zbl 1257.35143 |
| [60] | Langa, J. A.; Real, J.; Simon, J., Existence and regularity of the pressure for the stochastic Navier-Stokes equations. Appl. Math. Optim., 195-210 (2003) · Zbl 1049.60058 |
| [61] | Li, Y.-H.; Ding, S.-J.; Huang, M.-X., Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete Contin. Dyn. Syst., Ser. B, 1507-1523 (2016) · Zbl 1346.76195 |
| [62] | Li, Y.-H.; Huang, M.-X., Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Z. Angew. Math. Phys. (2018), 18 pp. · Zbl 1394.35362 |
| [63] | Liu, W.; Röckner, M., Stochastic Partial Differential Equations: an Introduction (2015), Springer: Springer Cham, pp. vi+266 · Zbl 1361.60002 |
| [64] | Lowengrub, J.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A, 2617-2654 (1998) · Zbl 0927.76007 |
| [65] | Orrieri, C.; Rocca, E.; Scarpa, L., Optimal control of stochastic phase-field models related to tumor growth. ESAIM Control Optim. Calc. Var., 104 (2020), 46 pp. · Zbl 1459.35415 |
| [66] | Orrieri, C.; Scarpa, L., Singular stochastic Allen-Cahn equations with dynamic boundary conditions. J. Differ. Equ., 8, 4624-4667 (2019) · Zbl 1461.35255 |
| [67] | Pardoux, E., Equations aux derivées partielles stochastiques nonlinéaires monotones (1975), Université Paris XI, PhD thesis · Zbl 0363.60041 |
| [68] | Röckner, M.; Schmuland, B.; Zhang, X., Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions. Condens. Matter Phys., 247-259 (2008) |
| [69] | Rubinstein, J.; Sternberg, P., Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math., 249-264 (1992) · Zbl 0763.35051 |
| [70] | Ryser, M.; Nigam, N.; Tupper, P. F., On the well-posedness of the stochastic Allen-Cahn equation in two dimensions. J. Comput. Phys., 6, 2537-2550 (2012) · Zbl 1242.65013 |
| [71] | Scarpa, L., The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential. Nonlinearity, 3813 (2021), 46 pp. · Zbl 1467.35199 |
| [72] | Scarpa, L., Analysis and optimal velocity control of a stochastic convective Cahn-Hilliard equation. J. Nonlinear Sci., 2, 45 (2021), 57 pp. · Zbl 1462.35478 |
| [73] | Scarpa, L., On the stochastic Cahn-Hilliard equation with a singular double-well potential. Nonlinear Anal., 102-133 (2018) · Zbl 1390.35122 |
| [74] | Scarpa, L., Optimal distributed control of a stochastic Cahn-Hilliard equation. SIAM J. Control Optim., 5, 3571-3602 (2019) · Zbl 1425.35090 |
| [75] | Scarpa, L., The stochastic viscous Cahn-Hilliard equation: well-posedness, regularity and vanishing viscosity limit. Appl. Math. Optim., 1, 487-533 (2021) · Zbl 1470.35452 |
| [76] | Simon, J., Compact sets in the space \(L^p(0, T; B)\). Ann. Mat. Pura Appl. (4), 65-96 (1987) · Zbl 0629.46031 |
| [77] | Tachim Medjo, T., On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete Contin. Dyn. Syst., 1, 395-430 (2019) · Zbl 1401.35252 |
| [78] | Tachim Medjo, T., On the existence and uniqueness of solution to a stochastic 2D Allen-Cahn-Navier-Stokes model. Stoch. Dyn. (2019), 28 pp. · Zbl 1410.35295 |
| [79] | Tachim Medjo, T., On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model. J. Differ. Equ., 1028-1054 (2017) · Zbl 1365.35235 |
| [80] | Tachim Medjo, T., On the weak solutions to a stochastic two-phase flow model. Appl. Anal., 914-937 (2022) · Zbl 1485.35442 |
| [81] | Tachim Medjo, T., On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects. Discrete Contin. Dyn. Syst., Ser. B, 10, 5447-5485 (2021) · Zbl 1496.35466 |
| [82] | van der Vaart, A. W.; Wellner, J. A., Weak Convergence and Empirical Processes. Springer Series in Statistics. With Applications to Statistics (1996), Springer-Verlag: Springer-Verlag New York, xvi+508 · Zbl 0862.60002 |
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