Attractors for a Navier–Stokes–Allen–Cahn system with unmatched densities

Chunyou Sun; Junyan Tan; Chunyou Sun; Junyan Tan

doi:10.3934/cam.2025010

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 237-262. doi: 10.3934/cam.2025010

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Research article

Attractors for a Navier–Stokes–Allen–Cahn system with unmatched densities

Chunyou Sun ^1,2,
Junyan Tan ^{1
,
,}

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China
2.
School of Mathematics and Statistics, Donghua University, Shanghai, 201620, P.R. China

Received: 26 January 2025 Revised: 04 March 2025 Accepted: 07 March 2025 Published: 14 March 2025
35B40, 35Q35, 35K61, 76T06

This paper investigates the long-time behavior for a Navier–Stokes–Allen–Cahn system, a diffuse interface model for two-phase incompressible flows with unmatched densities, non-constant viscosities, and a singular Flory–Huggins potential. First, we establish the dissipativity of strong solutions via some a priori estimates. Then, we demonstrate the regular-continuity of the semigroup, which allows us to prove the existence of the global attractor in the strong solutions space.
- two phase flow,
- global attractor,
- Navier–Stokes–Allen–Cahn system,
- dissipativity,
- singular Flory–Huggins potential
Citation: Chunyou Sun, Junyan Tan. Attractors for a Navier–Stokes–Allen–Cahn system with unmatched densities[J]. Communications in Analysis and Mechanics, 2025, 17(1): 237-262. doi: 10.3934/cam.2025010

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Abstract

This paper investigates the long-time behavior for a Navier–Stokes–Allen–Cahn system, a diffuse interface model for two-phase incompressible flows with unmatched densities, non-constant viscosities, and a singular Flory–Huggins potential. First, we establish the dissipativity of strong solutions via some a priori estimates. Then, we demonstrate the regular-continuity of the semigroup, which allows us to prove the existence of the global attractor in the strong solutions space.

References

[1]	S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085–1095. https://doi.org/10.1016/0001-6160(79)90196-2 doi: 10.1016/0001-6160(79)90196-2
[2]	D. T. Wasan, Interfacial transport processes and rheology: structure and dynamics of thin liquid films, Chem. Eng. Educ., 26 (1992), 104–112.
[3]	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. https://doi.org/10.1007/s00205-008-0160-2 doi: 10.1007/s00205-008-0160-2
[4]	D. M. Anderson, G. B. McFadden, A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139–165. https://doi.org/10.1146/annurev.fluid.30.1.139 doi: 10.1146/annurev.fluid.30.1.139
[5]	J. J. Feng, C. Liu, J. Shen, P. Yue, An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges, In: Modeling of Soft Matter, New York: Springer, 2005, 1–26. https://doi.org/10.1007/0-387-32153-5_1
[6]	J. Shen, X. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32 (2010), 1159–1179. https://doi.org/10.1137/09075860X doi: 10.1137/09075860X
[7]	J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A, 454 (1998), 2617–2654. https://doi.org/10.1098/rspa.1998.0273 doi: 10.1098/rspa.1998.0273
[8]	X. Chen, D. Hilhorst, E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces and Free Bound., 12 (2010), 527–549. https://doi.org/10.4171/IFB/244 doi: 10.4171/IFB/244
[9]	J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795–801. https://doi.org/10.1016/0001-6160(61)90182-1 doi: 10.1016/0001-6160(61)90182-1
[10]	Z. Han, R. Xu, Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity, Appl. Math. Lett., 160 (2025), 109290. https://doi.org/10.1016/j.aml.2024.109290 doi: 10.1016/j.aml.2024.109290
[11]	R. Xu, Y. Yang, B. Liu, J. Shen, S. Huang, Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015), 955–976. https://doi.org/10.1007/s00033-014-0459-9 doi: 10.1007/s00033-014-0459-9
[12]	Z. Han, R. Xu, Y. Yang, The qualitative behavior for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity, Discrete Contin. Dyn. Syst. -S, 16 (2023), 3131–3145. https://doi.org/10.3934/dcdss.2023141 doi: 10.3934/dcdss.2023141
[13]	R. Xu, Y. Luo, J. Shen, S. Huang, Global existence and blow up for damped generalized Boussinesq equation, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 251–262. https://doi.org/10.1007/s10255-017-0655-4 doi: 10.1007/s10255-017-0655-4
[14]	R. Chen, Z. Liang, D. Wang, R. Xu, On the inviscid limit of the compressible Navier-Stokes equations near Onsager's regularity in bounded domains, Sci. China Math., 67 (2024), 1–22. https://doi.org/10.1007/s11425-022-2085-3 doi: 10.1007/s11425-022-2085-3
[15]	R. Hosek, V. Macha, Weak-strong uniqueness for Navier-Stokes/Allen-Cahn system, Czechoslovak Math. J., 69 (2019), 837–851. https://doi.org/10.21136/CMJ.2019.0520-17 doi: 10.21136/CMJ.2019.0520-17
[16]	H. Wu, Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect, Eur. J. Appl. Math., 28 (2017), 380–434. https://doi.org/10.1017/S0956792516000322 doi: 10.1017/S0956792516000322
[17]	X. Xu, L. Zhao, C. Liu, Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations, SIAM J. Math. Anal., 41 (2010), 2246–2282. https://doi.org/10.1137/090754698 doi: 10.1137/090754698
[18]	Y. Chen, V. D. Radulescu, R. Xu, High energy blowup and blowup time for a class of semilinear parabolic equations with singular potential on manifolds with conical singularities, Commun. Math. Sci., 21 (2023), 25–63. https://doi.org/10.4310/CMS.2023.v21.n1.a2 doi: 10.4310/CMS.2023.v21.n1.a2
[19]	Q. Lin, R. Xu, Global well-posedness of the variable-order fractional wave equation with variable exponent nonlinearity, J. Lond. Math. Soc., 111 (2025), e70091. https://doi.org/10.1112/jlms.70091 doi: 10.1112/jlms.70091
[20]	M. Giga, A. Kirshtein, C. Liu, Variational modeling and complex fluids, In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Cham: Springer, 2018, 73–113. https://doi.org/10.1007/978-3-319-13344-7_2
[21]	J. Jiang, Y. Li, C. Liu, Two-phase incompressible flows with variable density: an energetic variational approach, Discrete Contin. Dyn. Syst., 37 (2017), 3243–3284. https://doi.org/10.3934/dcds.2017138 doi: 10.3934/dcds.2017138
[22]	C. Liu, J. Shen, X. Yang, Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density, J. Sci. Comput., 62 (2015), 601–622. https://doi.org/10.1007/s10915-014-9867-4 doi: 10.1007/s10915-014-9867-4
[23]	S. Chen, Y. Salmaniw, R. Xu, Global existence for a singular Gierer-Meinhardt system, J. Differ. Equ., 262 (2017), 2940–2960. https://doi.org/10.1016/j.jde.2016.11.022 doi: 10.1016/j.jde.2016.11.022
[24]	S. Chen, Y. Salmaniw, R. Xu, Bounded solutions to a singular parabolic system, J. Math. Anal. Appl., 455 (2017), 936–978. https://doi.org/10.1016/j.jmaa.2017.06.012 doi: 10.1016/j.jmaa.2017.06.012
[25]	R. Xu, W. Lian, Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321–356. https://doi.org/10.1007/s11425-017-9280-x doi: 10.1007/s11425-017-9280-x
[26]	R. Xu, J. Liu, Y. Niu, S. Chen, Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term, Bound. Value Probl., 2013 (2013), 1–13. https://doi.org/10.1186/1687-2770-2013-42 doi: 10.1186/1687-2770-2013-42
[27]	R. Xu, J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732–2763. https://doi.org/10.1016/j.jfa.2013.03.010 doi: 10.1016/j.jfa.2013.03.010
[28]	R. Xu, Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations'' [J. Func. Anal. 264 (2013) 2732–2763], J. Funct. Anal., 270 (2016), 4039–4041. https://doi.org/10.1016/j.jfa.2016.02.026 doi: 10.1016/j.jfa.2016.02.026
[29]	A. Giorgini, A. Miranville, R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535–2574. https://doi.org/10.1137/18M1223459 doi: 10.1137/18M1223459
[30]	A. Giorgini, R. Temam, Attractors for the Navier-Stokes-Cahn-Hilliard system, Discrete Contin. Dyn. Syst. -S, 15 (2022), 2249–2274. https://doi.org/10.3934/dcdss.2022118 doi: 10.3934/dcdss.2022118
[31]	A. Giorgini, M. Grasselli, H. Wu, On the mass-conserving Allen-Cahn approximation for incompressible binary fluids, J. Funct. Anal., 283 (2022), 109631. https://doi.org/10.1016/j.jfa.2022.109631 doi: 10.1016/j.jfa.2022.109631
[32]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, New York: Springer-Verlag, 1997. https://doi.org/10.1007/978-1-4612-0645-3
[33]	A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 95, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. https://doi.org/10.1137/1.9781611975925

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