August  2022, 15(8): 2345-2389. doi: 10.3934/dcdss.2022102

Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction

1. 

School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, 200433 Shanghai, China

2. 

School of Mathematical Sciences, Fudan University, 200433 Shanghai, China

* Corresponding author: Hao Wu

Dedicated to Prof. Maurizio Grasselli on the occasion of his 60th birthday, with admiration and best wishes.

Received  December 2021 Published  August 2022 Early access  April 2022

Fund Project: The first author was partially supported by NSF of China 12071084.

We study a hydrodynamic phase-field system modeling the deformation of functionalized membranes in incompressible viscous fluids. The governing PDE system consists of the Navier–Stokes equations coupled with a convective sixth-order Cahn–Hilliard type equation driven by the functionalized Cahn–Hilliard free energy, which describes the phase separation process in mixtures with an amphiphilic structure. In the three dimensional case, we prove existence of global weak solutions provided that the initial total energy is finite. Then we establish uniqueness of weak solutions under suitable regularity assumptions that are only imposed on the velocity field or its gradient. Next, we prove existence and uniqueness of local strong solutions for arbitrary regular initial data and derive some blow-up criteria. Finally, we show the eventual regularity of global weak solutions for large time. The results are obtained in a general setting with variable fluid viscosity and diffusion mobility.

Citation: Hao Wu, Yuchen Yang. Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2345-2389. doi: 10.3934/dcdss.2022102
References:
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S. Aland, Phase field models for two-phase flow with surfactants and biomembranes, Transport Processes at Fluidic Interfaces, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, (2017), 271–290.

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S. AlandS. EgererJ. Lowengrub and A. Voigt, Diffuse interface models of locally inextensible vesicles in a viscous fluid, J. Comput. Phys., 277 (2014), 32-47.  doi: 10.1016/j.jcp.2014.08.016.

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L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier–Stokes equations, Differ. Integral Equ., 15 (2002), 1129-1137. 

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F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Anal., 20 (1999), 175-212. 

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J. Brannick, A. Kirshtein and C. Liu, Dynamics of multi-component flows: Diffusive interface methods with energetic variational approaches, Reference Module in Materials Science and Materials Engineering, Elsevier, Oxford, (2016), 1–7.

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F. Campelo and A. Hernández-Machado, Dynamic model and stationary shapes of fluid vesicle, Eur. Phys. J. E, 20 (2006), 37-45. 

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P. Canham, The minimum energy of bending as a possible explanation of the bioconcave shape of the human red blood cell, J. Theoret. Biol., 26 (1970), 61-81. 

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K. ChengC. WangS. Wise and Z. Yuan, Global-in-time Gevrey regularity solutions for the functionalized Cahn–Hilliard equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2211-2229.  doi: 10.3934/dcdss.2020186.

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B. Climent-Ezquerra and F. Guillén-González, Long-time behavior of a Cahn–Hilliard–Navier–Stokes vesicle-fluid interaction model, Trends in Differential Equations and Applications, SEMA SIMAI Springer Ser., Springer, 8 (2016), 125-145.  doi: 10.1007/978-3-319-32013-7_8.

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B. Climent-Ezquerra and F. Guillén-González, Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model, Z. Angew. Math. Phys., 70 (2019), Paper No. 125, 27 pp. doi: 10.1007/s00033-019-1168-1.

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S. DaiQ. Liu and K. Promislow, Weak solutions for the functionalized Cahn–Hilliard equation with degenerate mobility, Appl. Anal., 100 (2021), 1-16.  doi: 10.1080/00036811.2019.1585536.

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S. Dai and K. Promislow, Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation, Proc. Roy. Soc. A, 469 (2013), 20120505, 20 pp. doi: 10.1098/rspa.2012.0505.

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Q. DuM.-L. Li and C. Liu, Analysis of a phase field Navier–Stokes vesicle-fluid interaction model, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 539-556.  doi: 10.3934/dcdsb.2007.8.539.

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Q. DuC. Liu and X.-Q. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys., 198 (2004), 450-468.  doi: 10.1016/j.jcp.2004.01.029.

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Q. DuC. Liu and X.-Q. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, J. Comput. Phys., 212 (2006), 757-777.  doi: 10.1016/j.jcp.2005.07.020.

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[28]

N. GavishG. HayrapetyanK. Promislow and L. Yang, Curvature driven flow of bi-layer interfaces, Phys. D, 240 (2011), 675-693. 

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N. GavishJ. JonesZ. XuA. Christlieb and K. Promislow, Variational models of network formation and ion transport: Applications to perfluorosulfonate ionomer membranes, Polymers, 4 (2012), 630-655. 

[30]

M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 73–113. doi: 10.1007/978-3-319-13344-7_2.

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A. Giorgini, Well-posedness of a diffuse interface model for Hele–Shaw flows, J. Math. Fluid Mech., 22 (2020), Paper No. 5, 36 pp. doi: 10.1007/s00021-019-0467-9.

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A. Giorgini, M. Grasselli and H. Wu, On the mass-conserving Allen–Cahn approximation for incompressible binary fluids, preprint, (2020).

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A. GiorginiA. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

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R. Hošek and and V. Mácha, Weak-strong uniqueness for Navier–Stokes/Allen–Cahn system, Czechoslovak Math. J., 69 (2019), 837-851.  doi: 10.21136/CMJ.2019.0520-17.

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J. JiangY.-H. Li and C. Liu, Two-phase incompressible flows with variable density: An energetic variational approach, Discrete Contin. Dyn. Syst., 37 (2017), 3243-3284.  doi: 10.3934/dcds.2017138.

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Y.-H. LiS.-J. Ding and M.-X. Huang, Blow-up criterion for an incompressible Navier–Stokes/Allen–Cahn system with different densities, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1507-1523.  doi: 10.3934/dcdsb.2016009.

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show all references

References:
[1]

H. Abels, On a diffuse interface model for two-phase flows of viscous incompressible fluids with matched densities, Arch. Rational Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.

[2]

S. Aland, Phase field models for two-phase flow with surfactants and biomembranes, Transport Processes at Fluidic Interfaces, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, (2017), 271–290.

[3]

S. AlandS. EgererJ. Lowengrub and A. Voigt, Diffuse interface models of locally inextensible vesicles in a viscous fluid, J. Comput. Phys., 277 (2014), 32-47.  doi: 10.1016/j.jcp.2014.08.016.

[4]

L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier–Stokes equations, Differ. Integral Equ., 15 (2002), 1129-1137. 

[5]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Anal., 20 (1999), 175-212. 

[6]

J. Brannick, A. Kirshtein and C. Liu, Dynamics of multi-component flows: Diffusive interface methods with energetic variational approaches, Reference Module in Materials Science and Materials Engineering, Elsevier, Oxford, (2016), 1–7.

[7]

F. Campelo and A. Hernández-Machado, Dynamic model and stationary shapes of fluid vesicle, Eur. Phys. J. E, 20 (2006), 37-45. 

[8]

P. Canham, The minimum energy of bending as a possible explanation of the bioconcave shape of the human red blood cell, J. Theoret. Biol., 26 (1970), 61-81. 

[9]

K. ChengC. WangS. Wise and Z. Yuan, Global-in-time Gevrey regularity solutions for the functionalized Cahn–Hilliard equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2211-2229.  doi: 10.3934/dcdss.2020186.

[10]

B. Climent-Ezquerra and F. Guillén-González, Long-time behavior of a Cahn–Hilliard–Navier–Stokes vesicle-fluid interaction model, Trends in Differential Equations and Applications, SEMA SIMAI Springer Ser., Springer, 8 (2016), 125-145.  doi: 10.1007/978-3-319-32013-7_8.

[11]

B. Climent-Ezquerra and F. Guillén-González, Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model, Z. Angew. Math. Phys., 70 (2019), Paper No. 125, 27 pp. doi: 10.1007/s00033-019-1168-1.

[12]

P. Colli and t">P. Laurenct, A phase-field approximation of the Willmore flow with volume constraints, Interfaces Free Bound., 13 (2011), 341-351.  doi: 10.4171/IFB/261.

[13]

P. Colli and t">P. Laurenct, A phase-field approximation of the Willmore flow with volume and area constraints, SIAM J. Math. Anal., 44 (2012), 3734-3754.  doi: 10.1137/120874126.

[14]

S. Dai, Q. Liu, T. Luong and K. Promislow, On nonnegative solutions for the functionalized Cahn–Hilliard equation with degenerate mobility, Results Appl. Math., 12 (2021), 100195, 13 pp. doi: 10.1016/j.rinam.2021.100195.

[15]

S. DaiQ. Liu and K. Promislow, Weak solutions for the functionalized Cahn–Hilliard equation with degenerate mobility, Appl. Anal., 100 (2021), 1-16.  doi: 10.1080/00036811.2019.1585536.

[16]

S. Dai and K. Promislow, Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation, Proc. Roy. Soc. A, 469 (2013), 20120505, 20 pp. doi: 10.1098/rspa.2012.0505.

[17]

S. Dai and K. Promislow, Competitive geometric evolution of amphiphilic interfaces, SIAM J. Math. Anal., 47 (2015), 347-380.  doi: 10.1137/130941432.

[18]

Q. DuM.-L. Li and C. Liu, Analysis of a phase field Navier–Stokes vesicle-fluid interaction model, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 539-556.  doi: 10.3934/dcdsb.2007.8.539.

[19]

Q. DuC. LiuR. Ryham and X.-Q. Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18 (2005), 1249-1267.  doi: 10.1088/0951-7715/18/3/016.

[20]

Q. DuC. LiuR. Ryham and X.-Q. Wang, Phase field modeling of the spontaneous curvature effect in cell membranes, Commun. Pure Appl. Anal., 4 (2005), 537-548. 

[21]

Q. DuC. LiuR. Ryham and X.-Q. Wang, Energetic variational approaches in modeling vesicle and fluid interactions, Phys. D, 238 (2009), 923-930.  doi: 10.1016/j.physd.2009.02.015.

[22]

Q. DuC. Liu and X.-Q. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys., 198 (2004), 450-468.  doi: 10.1016/j.jcp.2004.01.029.

[23]

Q. DuC. Liu and X.-Q. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, J. Comput. Phys., 212 (2006), 757-777.  doi: 10.1016/j.jcp.2005.07.020.

[24]

A. P. Entringer and J. L. Boldrini, A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 397-422.  doi: 10.3934/dcdsb.2015.20.397.

[25]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[26]

G. P. Galdi, An introduction to the Navier–Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, (2000), 1–70.

[27]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady State Problems, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[28]

N. GavishG. HayrapetyanK. Promislow and L. Yang, Curvature driven flow of bi-layer interfaces, Phys. D, 240 (2011), 675-693. 

[29]

N. GavishJ. JonesZ. XuA. Christlieb and K. Promislow, Variational models of network formation and ion transport: Applications to perfluorosulfonate ionomer membranes, Polymers, 4 (2012), 630-655. 

[30]

M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 73–113. doi: 10.1007/978-3-319-13344-7_2.

[31]

A. Giorgini, Well-posedness of a diffuse interface model for Hele–Shaw flows, J. Math. Fluid Mech., 22 (2020), Paper No. 5, 36 pp. doi: 10.1007/s00021-019-0467-9.

[32]

A. Giorgini, M. Grasselli and H. Wu, On the mass-conserving Allen–Cahn approximation for incompressible binary fluids, preprint, (2020).

[33]

A. GiorginiA. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[34]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsions and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335. 

[35]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. 

[36]

W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703. 

[37]

R. Hošek and and V. Mácha, Weak-strong uniqueness for Navier–Stokes/Allen–Cahn system, Czechoslovak Math. J., 69 (2019), 837-851.  doi: 10.21136/CMJ.2019.0520-17.

[38]

Y. HyonD. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle, Discrete Contin. Dyn. Syst., 26 (2010), 1291-1304.  doi: 10.3934/dcds.2010.26.1291.

[39]

J. JiangY.-H. Li and C. Liu, Two-phase incompressible flows with variable density: An energetic variational approach, Discrete Contin. Dyn. Syst., 37 (2017), 3243-3284.  doi: 10.3934/dcds.2017138.

[40]

N. Kajiwara, Strong well-posedness for the phase-field Navier–Stokes equations in the maximal regularity class, Commun. Math. Sci., 16 (2018), 239-250.  doi: 10.4310/CMS.2018.v16.n1.a11.

[41]

Y.-H. LiS.-J. Ding and M.-X. Huang, Blow-up criterion for an incompressible Navier–Stokes/Allen–Cahn system with different densities, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1507-1523.  doi: 10.3934/dcdsb.2016009.

[42]

F.-H. Lin and C. Liu, Nonparabolic dissipative system modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[43]

C. Liu and H. Wu, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis, Arch. Rational Mech. Anal., 233 (2019), 167-247.  doi: 10.1007/s00205-019-01356-x.

[44]

Y.-N. LiuT. Takahashi and M. Tucsnak, Strong solution for a phase field Navier–Stokes vesicle fluid interaction model, J. Math. Fluid Mech., 14 (2012), 177-195.  doi: 10.1007/s00021-011-0059-9.

[45]

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