pith. sign in

arxiv: 2507.01618 · v4 · submitted 2025-07-02 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

A Thermodynamically Consistent Free Boundary Model for Two-Phase Flows in an Evolving Domain with Bulk-Surface Interaction

Patrik Knopf , Yadong Liu This is my paper

Pith reviewed 2026-05-19 06:40 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords two-phase flowevolving domainbulk-surface interactionthermodynamic consistencyphase fieldCahn-HilliardNavier-Stokesfree boundary
0
0 comments X

The pith

The paper derives a thermodynamically consistent free boundary model for two-phase flows that includes bulk-surface material interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a mathematical framework for describing two-phase fluid mixtures in a domain whose boundary moves according to the flow velocity. Phase field descriptions are used both in the volume and on the surface to track the two components, with coupling terms that permit material to move from one to the other. The equations are obtained by enforcing local mass conservation and thermodynamic consistency through two derivation routes. The resulting system includes Navier-Stokes flow, convective Cahn-Hilliard evolution for the phases, and a slip condition that makes contact angles change with the dynamics.

Core claim

We derive from local mass balance a system of Navier-Stokes equations in the bulk coupled to convective Cahn-Hilliard equations on the bulk and surface. The model is thermodynamically consistent, satisfies local mass balance, allows material transfer between bulk and surface, produces variable contact angles, and recovers previous models by dropping dynamic boundary conditions or fixing the domain boundary.

What carries the argument

The coupled bulk-surface convective Cahn-Hilliard system with interaction terms designed for thermodynamic consistency, together with the generalized Navier slip boundary condition.

If this is right

  • Local mass balance is satisfied for the components.
  • Material can transfer between the bulk fluid and the surface.
  • Contact angles at the boundary vary dynamically.
  • Earlier models without dynamic boundaries or with fixed domains emerge as special cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar derivation techniques may apply to other systems involving interfaces with mass exchange, such as in multiphase porous media.
  • Testing the model against experiments with measurable surface adsorption rates could validate the transfer terms.

Load-bearing premise

The interaction terms between bulk and surface phase fields are assumed to take a form that guarantees the required energy dissipation while maintaining consistency with mass balance.

What would settle it

A calculation or simulation demonstrating that the energy dissipation law fails to hold under material transfer between bulk and surface would disprove the thermodynamic consistency.

Figures

Figures reproduced from arXiv: 2507.01618 by Patrik Knopf, Yadong Liu.

Figure 1
Figure 1. Figure 1: Droplet on a moving surface. (Only the relevant part of the evolving domain Ω(·) and its boundary is displayed.) 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolving domain (e.g., describing a bacterium) with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We derive a thermodynamically consistent model, which describes the time evolution of a two-phase flow in an evolving domain. The movement of the free boundary of the domain is driven by the velocity field of the mixture in the bulk, which is determined by a Navier--Stokes equation. In order to take interactions between bulk and boundary into account, we further consider two materials on the boundary, which may be the same or different materials as those in the bulk. The bulk and the surface materials are represented by respective phase-fields, whose time evolution is described by a bulk-surface convective Cahn--Hilliard equation. This approach allows for a transfer of material between bulk and surface as well as variable contact angles between the diffuse interface in the bulk and the boundary of the domain. To provide a more accurate description of the corresponding contact line motion, we include a generalized Navier slip boundary condition on the velocity field. Based on local mass balance laws, we derive our model from scratch in two different ways: by the Lagrange Multiplier Approach and (in the case of matched densities and no mass flux between bulk and surface) by the Energetic Variational Approach. We further show that our model generalizes previous models from the literature, which can be recovered from our system by either dropping the dynamic boundary conditions or assuming a static boundary of the domain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a thermodynamically consistent free-boundary model for two-phase incompressible flows in an evolving domain. Bulk flow is governed by Navier–Stokes with a generalized Navier slip condition; materials in the bulk and on the surface are represented by phase fields whose evolution is given by a coupled bulk-surface convective Cahn–Hilliard system that permits mass transfer across the interface. The model is obtained from local mass balance via a Lagrange-multiplier formulation (general case) and an energetic variational formulation (matched densities, no mass flux). It recovers several earlier models by specializing the boundary conditions or fixing the domain.

Significance. If the energy-dissipation identity is established for the general case with nonzero mass flux and distinct densities, the work supplies a unified, first-principles framework that simultaneously incorporates material exchange, variable contact angles, and dynamic contact-line motion. Such a model is of clear interest for the mathematical analysis of diffuse-interface flows with evolving boundaries.

major comments (2)
  1. [§3] §3 (Lagrange-multiplier derivation): the integration-by-parts identities that close the energy dissipation law when the surface velocity differs from the bulk trace and the domain is moving rely on the precise functional form of the bulk-surface coupling terms. Please exhibit the explicit cancellation of the cross terms for arbitrary densities and nonzero mass flux; the current argument appears to invoke the matched-density simplification used in the energetic-variational section.
  2. [§4] §4 (energy law): the dissipation identity is stated to be non-positive, yet the proof sketch is given only under the assumption of matched densities. A separate verification for the general case (distinct densities, active mass transfer) is required, because the curvature and transport terms introduced by the evolving domain may otherwise produce uncontrolled contributions.
minor comments (2)
  1. [Introduction] The notation for the surface phase field and the precise definition of the mass-transfer flux should be introduced with a single, self-contained table or diagram in the introduction.
  2. [Introduction] Several references to prior phase-field models with dynamic boundary conditions are cited only by author-year; please add the specific equation numbers from those works that are recovered as special cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments below and will revise the manuscript to strengthen the presentation of the energy-dissipation law.

read point-by-point responses

  1. Referee: [§3] §3 (Lagrange-multiplier derivation): the integration-by-parts identities that close the energy dissipation law when the surface velocity differs from the bulk trace and the domain is moving rely on the precise functional form of the bulk-surface coupling terms. Please exhibit the explicit cancellation of the cross terms for arbitrary densities and nonzero mass flux; the current argument appears to invoke the matched-density simplification used in the energetic-variational section.

    Authors: We appreciate the referee highlighting this point. The Lagrange-multiplier derivation in §3 is constructed for the general case with arbitrary densities and nonzero mass flux between bulk and surface. The bulk-surface coupling terms are chosen precisely so that the cross terms cancel upon integration by parts, even when the surface velocity differs from the bulk trace and the domain evolves. To improve clarity and address the concern directly, we will add an explicit step-by-step computation of these cancellations in the revised manuscript, without invoking the matched-density assumption used later in the energetic-variational section. revision: yes

  2. Referee: [§4] §4 (energy law): the dissipation identity is stated to be non-positive, yet the proof sketch is given only under the assumption of matched densities. A separate verification for the general case (distinct densities, active mass transfer) is required, because the curvature and transport terms introduced by the evolving domain may otherwise produce uncontrolled contributions.

    Authors: We agree that the current sketch of the energy law focuses on the matched-density case. Although the Lagrange-multiplier formulation guarantees thermodynamic consistency for the general system, an explicit verification for distinct densities and active mass transfer is indeed desirable to control the additional curvature and transport terms arising from the moving domain. In the revised version we will supply a complete, self-contained proof of the dissipation identity in the general case, showing that all such terms cancel or are absorbed into the non-positive dissipation. revision: yes

Circularity Check

0 steps flagged

Derivation from local mass balance laws via Lagrange multipliers and energetic variational approach is self-contained

full rationale

The paper derives its system from scratch starting from local mass balance laws, employing two independent methods (Lagrange multiplier approach for the general case and energetic variational approach for matched densities with no mass flux). The bulk-surface coupling terms are constructed within this derivation to enforce thermodynamic consistency and mass conservation, rather than being fitted or renamed from the target result. The model generalizes prior literature by dropping dynamic boundary conditions or fixing the domain, providing independent content. Self-citations to phase-field models supply background but do not load-bear the central claims or force uniqueness via internal theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The model rests on standard continuum mechanics assumptions plus specific choices for the interaction potentials and mobility functions that are introduced to close the system while preserving thermodynamic consistency.

axioms (2)
  • domain assumption Local mass balance laws hold separately in the bulk and on the surface, with possible exchange terms between them.
    Invoked at the start of the derivation to obtain the evolution equations for the phase fields.
  • domain assumption The total energy functional includes bulk and surface contributions whose time derivative yields a non-positive dissipation.
    Central to both the Lagrange multiplier and energetic variational derivations.
invented entities (1)
  • Bulk-surface interaction terms in the phase-field equations no independent evidence
    purpose: To allow material transfer and enforce thermodynamic consistency across the interface
    Introduced as part of the model construction; no independent experimental evidence cited in abstract.

pith-pipeline@v0.9.0 · 5785 in / 1511 out tokens · 21911 ms · 2026-05-19T06:40:22.871072+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    Abels, D

    H. Abels, D. Depner, and H. Garcke. Existence of weak solutions for a diffuse interface model for two- phase flows of incompressible fluids with different densities. Journal of Mathematical Fluid Mechanics , 15(3):453–480, 2013

    work page 2013

  2. [2]

    Abels, D

    H. Abels, D. Depner, and H. Garcke. On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility. Annales de l’Institut Henri Poincar´ e C, 30(6):1175–1190, 2013. 20

    work page 2013

  3. [3]

    Abels and H

    H. Abels and H. Garcke. Weak solutions and diffuse interface models for incompressible two-phase flows. In Handbook of mathematical analysis in mechanics of viscous fluids , pages 1267–1327. Springer, Cham, 2018

    work page 2018

  4. [4]

    Abels, H

    H. Abels, H. Garcke, and A. Giorgini. Global regularity and asymptotic stabilization for the incompressible Navier-Stokes-Cahn-Hilliard model with unmatched densities. Math. Ann., 389(2):1267–1321, 2024

    work page 2024

  5. [5]

    Abels, H

    H. Abels, H. Garcke, and G. Gr¨ un. Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. , 22(3):1150013, 40, 2012

    work page 2012

  6. [6]

    Abels, H

    H. Abels, H. Garcke, and A. Poiatti. Mathematical analysis of a diffuse interface model for multi-phase flows of incompressible viscous fluids with different densities. J. Math. Fluid Mech. , 26(2):Paper No. 29, 51, 2024

    work page 2024

  7. [7]

    Abels, H

    H. Abels, H. Garcke, and A. Poiatti. Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness. Calc. Var. Partial Differential Equations , 64(5):Paper No. 141, 41, 2025

    work page 2025

  8. [8]

    Abels, H

    H. Abels, H. Garcke, and J. Wittmann. Diffuse Interface Models for Two-Phase Flows with Phase Transition: Modeling and Existence of Weak Solutions. Preprint: arXiv:2505.05383 [math.AP], 2025

    work page arXiv 2025

  9. [9]

    Abels and J

    H. Abels and J. Weber. Local well-posedness of a quasi-incompressible two-phase flow. J. Evol. Equ. , 21(3):3477–3502, 2021

    work page 2021

  10. [10]

    G. L. Aki, W. Dreyer, J. Giesselmann, and C. Kraus. A quasi-incompressible diffuse interface model with phase transition. Math. Models Methods Appl. Sci. , 24(5):827–861, 2014

    work page 2014

  11. [11]

    Bachini, V

    E. Bachini, V. Krause, I. Nitschke, and A. Voigt. Derivation and simulation of a two-phase fluid deformable surface model. J. Fluid Mech. , 977:Paper No. A41, 37, 2023

    work page 2023

  12. [12]

    Bachini, V

    E. Bachini, V. Krause, and A. Voigt. The interplay of geometry and coarsening in multicomponent lipid vesicles under the influence of hydrodynamics. Physics of Fluids , 35(4):042102, 04 2023

    work page 2023

  13. [13]

    B¨ ansch, K

    E. B¨ ansch, K. Deckelnick, H. Garcke, and P. Pozzi.Interfaces: Modeling, Analysis, Numerics. Oberwolfach Seminars. Birkh¨ auser Cham, 2023

    work page 2023

  14. [14]

    Bao and H

    X. Bao and H. Zhang. Numerical approximations and error analysis of the Cahn–Hilliard equation with dynamic boundary conditions. Commun. Math. Sci. , 19(3):663–685, 2021

    work page 2021

  15. [15]

    Bao and H

    X. Bao and H. Zhang. Numerical approximations and error analysis of the Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions. J. Sci. Comput. , 87(3):Paper No. 72, 32, 2021

    work page 2021

  16. [16]

    J. W. Barrett, H. Garcke, and R. N¨ urnberg. Finite element approximation for the dynamics of fluidic two-phase biomembranes. ESAIM Math. Model. Numer. Anal. , 51(6):2319–2366, 2017

    work page 2017

  17. [17]

    J. W. Barrett, H. Garcke, and R. N¨ urnberg. Parametric finite element approximations of curvature-driven interface evolutions. In Geometric partial differential equations. Part I , volume 21 of Handb. Numer. Anal., pages 275–423. Elsevier/North-Holland, Amsterdam, [2020] ©2020

    work page 2020

  18. [18]

    Binder and H

    K. Binder and H. L. Frisch. Dynamics of surface enrichment: A theory based on the Kawasaki spin- exchange model in the presence of a wall. Z. Phys. B , 84:403—-418, 1991

    work page 1991

  19. [19]

    F. Boyer. A theoretical and numerical model for the study of incompressible mixture flows. Comput. & Fluids, 31(1):41–68, 2002

    work page 2002

  20. [20]

    Bullerjahn

    N. Bullerjahn. Error estimates for full discretization by an almost mass conservation technique for Cahn– Hilliard systems with dynamic boundary conditions. Preprint: arXiv:2502.03847 [math.AP], 2025

    work page arXiv 2025

  21. [21]

    Bullerjahn and B

    N. Bullerjahn and B. Kov´ acs. Error estimates for full discretization of cahn–hilliard equation with dynamic boundary conditions. IMA Journal of Numerical Analysis , page draf009, 04 2025

    work page 2025

  22. [22]

    Caetano and C

    D. Caetano and C. M. Elliott. Cahn-Hilliard equations on an evolving surface. European J. Appl. Math. , 32(5):937–1000, 2021

    work page 2021

  23. [23]

    Caetano, C

    D. Caetano, C. M. Elliott, M. Grasselli, and A. Poiatti. Regularization and separation for evolving surface Cahn-Hilliard equations. SIAM J. Math. Anal. , 55(6):6625–6675, 2023

    work page 2023

  24. [24]

    Cherfils, E

    L. Cherfils, E. Feireisl, M. Mich´ alek, A. Miranville, M. Petcu, and D. Praˇ z´ ak. The compressible Navier- Stokes-Cahn-Hilliard equations with dynamic boundary conditions. Math. Models Methods Appl. Sci. , 29(14):2557–2584, 2019

    work page 2019

  25. [25]

    Colli, T

    P. Colli, T. Fukao, and L. Scarpa. The Cahn–Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity. SIAM J. Math. Anal. , 54(3):3292–3315, 2022

    work page 2022

  26. [26]

    Colli, T

    P. Colli, T. Fukao, and L. Scarpa. A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials. J. Evol. Equ. , 22(4):Paper No. 89, 31, 2022

    work page 2022

  27. [27]

    Colli, T

    P. Colli, T. Fukao, and H. Wu. On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials. Math. Nachr., 293(11):2051–2081, 2020

    work page 2051

  28. [28]

    Colli, G

    P. Colli, G. Gilardi, and J. Sprekels. On a Cahn–Hilliard system with convection and dynamic boundary conditions. Ann. Mat. Pura Appl. (4) , 197(5):1445–1475, 2018. 21

    work page 2018

  29. [29]

    Colli, G

    P. Colli, G. Gilardi, and J. Sprekels. Optimal velocity control of a convective Cahn–Hilliard system with double obstacles and dynamic boundary conditions: a ‘deep quench’ approach. J. Convex Anal. , 26(2):485–514, 2019

    work page 2019

  30. [30]

    Colli, P

    P. Colli, P. Knopf, G. Schimperna, and A. Signori. Two-phase flows through porous media described by a Cahn-Hilliard-Brinkman model with dynamic boundary conditions. J. Evol. Equ. , 24(4):Paper No. 85, 55, 2024

    work page 2024

  31. [31]

    H. Ding, P. D. Spelt, and C. Shu. Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. , 226(2):2078–2095, 2007

    work page 2078

  32. [32]

    Du and X

    Q. Du and X. Feng. The phase field method for geometric moving interfaces and their numerical approx- imations. In Geometric partial differential equations. Part I , volume 21 of Handb. Numer. Anal. , pages 425–508. Elsevier/North-Holland, Amsterdam, 2020

    work page 2020

  33. [33]

    C. M. Elliott and T. Sales. Navier–Stokes–Cahn–Hilliard equations on evolving surfaces. Interfaces Free Bound., online first, 2024

    work page 2024

  34. [34]

    Feireisl, H

    E. Feireisl, H. Petzeltov´ a, E. Rocca, and G. Schimperna. Analysis of a phase-field model for two-phase compressible fluids. Math. Models Methods Appl. Sci. , 20(7):1129–1160, 2010

    work page 2010

  35. [35]

    Fischer, P

    H. Fischer, P. Maass, and W. Dieterich. Novel Surface Modes in Spinodal Decomposition. Phys. Rev. Lett., 79:893–896, 1997

    work page 1997

  36. [36]

    Fischer, J

    H. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer, and D. Reinel. Time- dependent density functional theory and the kinetics of lattice gas systems in contact with a wall. J. Chem. Phys., 108(7):3028–3037, 1998

    work page 1998

  37. [37]

    Freist¨ uhler and M

    H. Freist¨ uhler and M. Kotschote. Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids. Archive for Rational Mechanics and Analysis , 224(1):1–20, 2017

    work page 2017

  38. [38]

    Fukao and H

    T. Fukao and H. Wu. Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn–Hilliard type with singular potential. Asymptot. Anal. , 124(3-4):303–341, 2021

    work page 2021

  39. [39]

    C. G. Gal. A Cahn–Hilliard model in bounded domains with permeable walls. Math. Methods Appl. Sci. , 29(17):2009–2036, 2006

    work page 2009

  40. [40]

    C. G. Gal, M. Grasselli, and A. Miranville. Cahn–Hilliard–Navier–Stokes systems with moving contact lines. Calc. Var. Partial Differential Equations , 55(3):Art. 50, 47, 2016

    work page 2016

  41. [41]

    C. G. Gal, M. Grasselli, and H. Wu. Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities. Arch. Ration. Mech. Anal., 234(1):1–56, 2019

    work page 2019

  42. [42]

    C. G. Gal, M. Lv, and H. Wu. On a thermodynamically consistent diffuse interface model for two-phase incompressible flows with non-matched densities: Dynamics of moving contact lines, surface diffusion, and mass transfer. Preprint on ResearchGate: https://doi.org/10.13140/RG.2.2.27875.73764, 2024

    work page doi:10.13140/rg.2.2.27875.73764 2024

  43. [43]

    Garcke and P

    H. Garcke and P. Knopf. Weak solutions of the Cahn–Hilliard system with dynamic boundary conditions: a gradient flow approach. SIAM J. Math. Anal. , 52(1):340–369, 2020

    work page 2020

  44. [44]

    Garcke, P

    H. Garcke, P. Knopf, and S. Yayla. Long-time dynamics of the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions. Nonlinear Anal., 215:Paper No. 112619, 44, 2022

    work page 2022

  45. [45]

    Garcke, K

    H. Garcke, K. F. Lam, and B. Stinner. Diffuse interface modelling of soluble surfactants in two-phase flow. Commun. Math. Sci. , 12(8):1475–1522, 2014

    work page 2014

  46. [46]

    M.-H. Giga, A. Kirshtein, and C. Liu. Variational modeling and complex fluids. In Handbook of mathe- matical analysis in mechanics of viscous fluids , pages 73–113. Springer, Cham, 2018

    work page 2018

  47. [47]

    Gilardi and J

    G. Gilardi and J. Sprekels. Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions. Nonlinear Anal., 178:1–31, 2019

    work page 2019

  48. [48]

    Giorgini

    A. Giorgini. Well-posedness of the two-dimensional Abels–Garcke–Gr¨ un model for two-phase flows with unmatched densities. Calculus of Variations and Partial Differential Equations , 60(3):1–40, 2021

    work page 2021

  49. [49]

    Giorgini

    A. Giorgini. Existence and stability of strong solutions to the Abels-Garcke-Gr¨ un model in three dimen- sions. Interfaces Free Bound., online first, 2022

    work page 2022

  50. [50]

    Giorgini and P

    A. Giorgini and P. Knopf. Two-phase flows with bulk-surface interaction: thermodynamically consistent Navier-Stokes-Cahn–Hilliard models with dynamic boundary conditions. J. Math. Fluid Mech. , 25(3):Pa- per No. 65, 44, 2023

    work page 2023

  51. [51]

    Giorgini, P

    A. Giorgini, P. Knopf, and J. Stange. Regularity of a bulk-surface Cahn-Hilliard model driven by Leray velocity fields. Preprint: arXiv:2506.18617 [math.AP]

    work page arXiv

  52. [52]

    G. R. Goldstein, A. Miranville, and G. Schimperna. A Cahn–Hilliard model in a domain with non- permeable walls. Phys. D, 240(8):754–766, 2011

    work page 2011

  53. [53]

    Gurtin, D

    M. Gurtin, D. Polignone, and J. Vi˜ nals. Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. , 6(6):815–831, 1996. 22

    work page 1996

  54. [54]

    Harder and B

    P. Harder and B. Kov´ acs. Error estimates for the Cahn–Hilliard equation with dynamic boundary condi- tions. IMA J. Numer. Anal. , 42(3):2589–2620, 2022

    work page 2022

  55. [55]

    Heida, J

    M. Heida, J. M´ alek, and K. Rajagopal. On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework. Zeitschrift f¨ ur angewandte Mathematik und Physik, 63(1):145–169, 2012

    work page 2012

  56. [56]

    P. C. Hohenberg and B. I. Halperin. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49:435–479, 1977

    work page 1977

  57. [57]

    Kenzler, F

    R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl, and W. Dietrich. Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions. Comp. Phys. Comm., 133:139–157, 2001

    work page 2001

  58. [58]

    Knopf and K

    P. Knopf and K. F. Lam. Convergence of a Robin boundary approximation for a Cahn–Hilliard system with dynamic boundary conditions. Nonlinearity, 33(8):4191–4235, 2020

    work page 2020

  59. [59]

    Knopf, K

    P. Knopf, K. F. Lam, C. Liu, and S. Metzger. Phase-field dynamics with transfer of materials: the Cahn-Hillard equation with reaction rate dependent dynamic boundary conditions. ESAIM Math. Model. Numer. Anal., 55(1):229–282, 2021

    work page 2021

  60. [60]

    Knopf and A

    P. Knopf and A. Signori. On the nonlocal Cahn–Hilliard equation with nonlocal dynamic boundary condition and boundary penalization. J. Differential Equations , 280:236–291, 2021

    work page 2021

  61. [61]

    Knopf and J

    P. Knopf and J. Stange. Well-posedness of a bulk-surface convective Cahn–Hilliard system with dynamic boundary conditions. NoDEA Nonlinear Differential Equations Appl. , 31(5):Paper No. 82, 48, 2024

    work page 2024

  62. [62]

    Knopf and J

    P. Knopf and J. Stange. Strong well-posedness and separation properties for a bulk-surface convective Cahn–Hilliard system with singular potentials. J. Differential Equations , 439:113408, 2025

    work page 2025

  63. [63]

    Liu and H

    C. Liu and H. Wu. An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal., 233(1):167– 247, 2019

    work page 2019

  64. [64]

    I.-S. Liu. Continuum mechanics. Advanced Texts in Physics. Springer-Verlag, Berlin, 2002

    work page 2002

  65. [65]

    Lowengrub and L

    J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. , 454(1978):2617–2654, 1998

    work page 1978

  66. [66]

    Lv and H

    M. Lv and H. Wu. On the nonlocal Cahn–Hilliard equation with nonlocal dynamic boundary condition and singular potential: well-posedness, regularity and asymptotic limits. Nonlinearity, 38(4):045017, 2025

    work page 2025

  67. [67]

    X. Meng, X. Bao, and Z. Zhang. Second order stabilized semi-implicit scheme for the Cahn–Hilliard model with dynamic boundary conditions. J. Comput. Appl. Math. , 428:Paper No. 115145, 22, 2023

    work page 2023

  68. [68]

    S. Metzger. An efficient and convergent finite element scheme for Cahn–Hilliard equations with dynamic boundary conditions. SIAM J. Numer. Anal. , 59(1):219–248, 2021

    work page 2021

  69. [69]

    S. Metzger. A convergent SAV scheme for Cahn–Hilliard equations with dynamic boundary conditions. IMA Journal of Numerical Analysis , page drac078, 01 2023

    work page 2023

  70. [70]

    Miranville

    A. Miranville. The Cahn–Hilliard equation: Recent advances and applications , volume 95 of CBMS-NSF Regional Conference Series in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019

    work page 2019

  71. [71]

    Miranville and H

    A. Miranville and H. Wu. Long-time behavior of the Cahn–Hilliard equation with dynamic boundary condition. J. Elliptic Parabol. Equ. , 6(1):283–309, 2020

    work page 2020

  72. [72]

    Pr¨ uss and G

    J. Pr¨ uss and G. Simonett.Moving interfaces and quasilinear parabolic evolution equations , volume 105 of Monographs in Mathematics. Birkh¨ auser/Springer, [Cham], 2016

    work page 2016

  73. [73]

    Qian, X.-P

    T. Qian, X.-P. Wang, and P. Sheng. A variational approach to moving contact line hydrodynamics. J. Fluid Mech., 564:333–360, 2006

    work page 2006

  74. [74]

    Seppecher

    P. Seppecher. Moving contact lines in the Cahn-Hilliard theory. International Journal of Engineering Science, 34(9):977–992, 1996

    work page 1996

  75. [75]

    J. Shen, X. Yang, and Q. Wang. Mass and volume conservation in phase field models for binary fluids. Communications in Computational Physics , 13(4):1045–1065, 2013

    work page 2013

  76. [76]

    Shokrpour Roudbari, G

    M. Shokrpour Roudbari, G. S ¸im¸ sek, E. H. van Brummelen, and K. G. van der Zee. Diffuse-interface two- phase flow models with different densities: a new quasi-incompressible form and a linear energy-stable method. Math. Models Methods Appl. Sci. , 28(4):733–770, 2018

    work page 2018

  77. [77]

    M. F. P. ten Eikelder, K. G. van der Zee, I. Akkerman, and D. Schillinger. A unified framework for Navier- Stokes Cahn-Hilliard models with non-matching densities. Math. Models Methods Appl. Sci. , 33(1):175– 221, 2023

    work page 2023

  78. [78]

    H. Wu. A review on the Cahn–Hilliard equation: classical results and recent advances in dynamic boundary conditions. Electron. Res. Arch., 30(8):2788–2832, 2022

    work page 2022

  79. [79]

    J. Yao, Z. Zhang, and W. Ren. Modelling moving contact lines on inextensible elastic sheets in two dimensions. J. Fluid Mech. , 955:Paper No. A25, 29, 2023. 23

    work page 2023