pith. sign in

arxiv: 2411.09455 · v2 · submitted 2024-11-14 · 🧮 math.AP

Local-in-time existence of strong solutions to a quasi-incompressible Cahn--Hilliard--Navier--Stokes system

Mingwen Fei , Xiang Fei , Daozhi Han , Yadong Liu This is my paper

Pith reviewed 2026-05-23 17:45 UTC · model grok-4.3

classification 🧮 math.AP
keywords quasi-incompressibleCahn-Hilliard-Navier-Stokesstrong solutionslocal existenceuniquenessmaximal regularitytwo-phase flowsBanach fixed point
0
0 comments X

The pith

The quasi-incompressible Cahn-Hilliard-Navier-Stokes system admits local-in-time unique strong solutions for two-phase flows with unmatched densities.

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

This paper proves the local existence and uniqueness of strong solutions to a quasi-incompressible version of the Cahn-Hilliard-Navier-Stokes equations. The model describes two-phase fluid flows where the two fluids have different densities, using the volume fraction difference as the order parameter and the mass-averaged velocity. Pressure enters the chemical potential equation due to the quasi-incompressibility. The proof combines the Banach fixed point theorem with maximal regularity estimates for the associated linear system. This result provides a rigorous basis for the short-time behavior of such coupled phase-field and fluid models.

Core claim

The paper establishes local existence and uniqueness of strong solutions to the quasi-incompressible Cahn-Hilliard-Navier-Stokes system by applying the Banach fixed point theorem to a suitable map derived from the maximal regularity theory of the linearized system.

What carries the argument

Banach fixed point theorem combined with maximal regularity theory for the linearized quasi-incompressible Cahn-Hilliard-Navier-Stokes system

If this is right

  • Strong solutions exist on a positive but possibly small time interval determined by the initial data.
  • The solutions are unique in the function spaces where the maximal regularity theory applies.
  • The quasi-incompressible structure incorporates pressure into the chemical potential equation.
  • The result covers two-phase flows with unmatched densities using volume fraction difference and mass-averaged velocity.

Where Pith is reading between the lines

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

  • Numerical approximations of the system could be justified rigorously for sufficiently short times.
  • The local theory might serve as a starting point for studying possible finite-time singularities or global existence under extra smallness conditions.
  • Related models with different velocity formulations or compressibility assumptions could be analyzed by similar fixed-point arguments.

Load-bearing premise

The initial data and parameters must lie in function spaces where maximal regularity applies to the linearized system and the fixed-point map contracts on a small time interval.

What would settle it

Initial data in the relevant spaces for which the contraction mapping fails to produce a fixed point on any positive time interval, or for which no strong solution exists locally.

read the original abstract

We analyze a quasi-incompressible Cahn--Hilliard--Navier--Stokes system (qCHNS) for two-phase flows with unmatched densities. The order parameter is the volume fraction difference of the two fluids, while mass-averaged velocity is adopted. This leads to a quasi-incompressible model where the pressure also enters the equation of the chemical potential. We establish local existence and uniqueness of strong solutions by the Banach fixed point theorem and the maximal regularity theory.

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

0 major / 3 minor

Summary. The manuscript establishes local-in-time existence and uniqueness of strong solutions to a quasi-incompressible Cahn-Hilliard-Navier-Stokes system modeling two-phase flows with unmatched densities. The order parameter is the volume-fraction difference and the velocity is mass-averaged, so that pressure appears in the chemical-potential equation. The proof proceeds by constructing a fixed-point map from the solution operator of the linearized system furnished by maximal regularity theory and showing that the map is a contraction on a sufficiently short time interval.

Significance. If the linear theory is correctly established in the chosen spaces, the result supplies a rigorous local well-posedness theory for a physically relevant quasi-incompressible model. The explicit appeal to maximal regularity and the Banach fixed-point theorem is a standard and appropriate route once the linear estimates are available; this constitutes a clear technical contribution.

minor comments (3)
  1. [Abstract] The abstract introduces the abbreviation qCHNS without spelling it out; define the acronym on first use.
  2. [Theorem 1.1 (or equivalent)] In the statement of the main existence theorem, list the precise function spaces for the initial data and the compatibility conditions required by the maximal-regularity framework.
  3. [Section 4 (fixed-point argument)] Verify that all constants appearing in the contraction estimate are independent of the small time interval T; if any dependence remains, state it explicitly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies the Banach fixed-point theorem to a contraction mapping constructed from the solution operator of the linearized system, whose maximal regularity is invoked as an external theorem in appropriate function spaces. This is a standard, non-circular route for local strong solutions of quasilinear parabolic systems; the initial data and small-time contraction are chosen to satisfy the hypotheses of those independent theorems rather than being defined in terms of the target existence result. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard functional-analytic tools rather than new axioms or fitted parameters; the abstract supplies no free parameters or invented entities.

axioms (2)
  • standard math Banach fixed point theorem applies in a suitable Banach space of solutions
    Invoked explicitly in the abstract to obtain the fixed-point solution.
  • domain assumption Maximal regularity estimates hold for the linearized quasi-incompressible system
    Required for the contraction mapping argument; standard in parabolic PDE theory but must be verified for this pressure-coupled system.

pith-pipeline@v0.9.0 · 5614 in / 1268 out tokens · 27709 ms · 2026-05-23T17:45:25.120488+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

47 extracted references · 47 canonical work pages

  1. [1]

    H. Abels, Diffuse Interface Models for Two-Phase Flows of Viscous Incom pressible Fluids, Lecture Notes 36/2007, Max Planck Institute for Mathematics in the Sciences, Leipz ig, Germany, 2007

    work page 2007

  2. [2]

    Abels, Existence of weak solutions for a diffuse interface model for v iscous, incompressible fluids with general densities, Commun

    H. Abels, Existence of weak solutions for a diffuse interface model for v iscous, incompressible fluids with general densities, Commun. Math. Phys. 289(2009), 45-73

    work page 2009

  3. [3]

    Abels, Strong well-posedness of a diffuse interface model for a visco us, quasi-incompressible two-phase flow , SIAM J

    H. Abels, Strong well-posedness of a diffuse interface model for a visco us, quasi-incompressible two-phase flow , SIAM J. Math. Anal. 44(2012), 316-340

    work page 2012

  4. [4]

    Abels, D

    H. Abels, D. Breit, Weak solutions for a non-Newtonian diffuse interface model wi th different densities , Nonlin- earity 29(2016), 3426-3453

    work page 2016

  5. [5]

    Abels, D

    H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for t wo-phase flows of incompressible fluids with different densities , J. Math. Fluid Mech. 15(2013), 453-480

    work page 2013

  6. [6]

    Abels, D

    H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire 30(2013),1175-1190

    work page 2013

  7. [7]

    Abels, H

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

    work page 2024

  8. [8]

    Abels, H

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

    work page 2012

  9. [9]

    Abels, J

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

    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(2014), 827-861

    work page 2014

  11. [11]

    Amann, Linear and Quasilinear Parabolic Problems , Volume 1: Abstract Linear Theory, Birkh¨ auser, Basel- Boston-Berlin, 1995

    H. Amann, Linear and Quasilinear Parabolic Problems , Volume 1: Abstract Linear Theory, Birkh¨ auser, Basel- Boston-Berlin, 1995

    work page 1995

  12. [12]

    Bergh, J

    J. Bergh, J. L¨ ofstr¨ om,Interpolation Spaces, Springer, Berlin-Heidelberg-New York, 1976

    work page 1976

  13. [13]

    Boyer, A theoretical and numerical model for the study of incompres sible mixture flows , Comput

    F. Boyer, A theoretical and numerical model for the study of incompres sible mixture flows , Comput. Fluids 31(2002), 41-68

    work page 2002

  14. [14]

    S. P. Chen, R. Triggiani, Proof of extensions of two conjectures on structural dampin g for elastic systems , Pacific J. Math. 136(1989), 15-55

    work page 1989

  15. [15]

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

    work page 2007

  16. [16]

    Frigeri, Global existence of weak solutions for a nonlocal model for t wo-phase flows of incompressible fluids with unmatched densities , Math

    S. Frigeri, Global existence of weak solutions for a nonlocal model for t wo-phase flows of incompressible fluids with unmatched densities , Math. Models Methods Appl. Sci. 26(2016), 1955-1993

    work page 2016

  17. [17]

    Frigeri, On a nonlocal Cahn-Hilliard/Navier-Stokes system with deg enerate mobility and singular potential for incompressible fluids with different densities , Ann

    S. Frigeri, On a nonlocal Cahn-Hilliard/Navier-Stokes system with deg enerate mobility and singular potential for incompressible fluids with different densities , Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire 38(2021),647-687

    work page 2021

  18. [18]

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

    work page 2019

  19. [19]

    G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-St okes Equations , Volume 1, Springer, Berlin-Heidelberg-New York, 1994

    work page 1994

  20. [20]

    Giorgini, Well-posedness of the two-dimensional Abels-Garcke-Gr¨ un model for two-phase flows with unmatched densities, Calc

    A. Giorgini, Well-posedness of the two-dimensional Abels-Garcke-Gr¨ un model for two-phase flows with unmatched densities, Calc. Var. Partial Differ. Equ. 60(2021), Paper No.100, 40p p

    work page 2021

  21. [21]

    Giorgini, Existence and stability of strong solutions to the Abels-Ga rcke-Gr¨ un model in three dimensions , Interfaces Free Bound

    A. Giorgini, Existence and stability of strong solutions to the Abels-Ga rcke-Gr¨ un model in three dimensions , Interfaces Free Bound. 24(2022), 565-608. 28 MINGWEN FEI, XIANG FEI, DAOZHI HAN, AND YADONG LIU

    work page 2022

  22. [22]

    Giorgini, R

    A. Giorgini, R. Temam, Weak and strong solutions to the nonhomogeneous incompress ible Navier-Stokes-Cahn- Hilliard system , J. Math. Pures Appl. 9(2020), 194-249

    work page 2020

  23. [23]

    Gomez, K

    H. Gomez, K. G. van der Zee, Computational phase-field modeling , in Encyclopedia of Computational Mechanics, 2nd ed., John Wiley & Sons, New York, 2017

    work page 2017

  24. [24]

    Gr¨ un, F

    G. Gr¨ un, F. Guill´ en-Gonz´ alez and S. Metzger,On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities , Commun. Comput. Phys. 19(2016), 1473-1502

    work page 2016

  25. [25]

    Z. L. Guo, Q. Chen, P. Lin, C. Liu and J. Lowengrub, Second order approximation for a quasi-incompressible Navier-Stokes Cahn-Hilliard system of two-phase flows with variable density , J. Comput. Phys, 448(2022), Paper No. 110727, 17pp

    work page 2022

  26. [26]

    M. E. 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(1996), 815-831

    work page 1996

  27. [27]

    Hanouzet, Applications bilin´ eaires compatibles avec un syst` eme ` a coefficients variables , Continuit´ e dans les espaces de Besov

    B. Hanouzet, Applications bilin´ eaires compatibles avec un syst` eme ` a coefficients variables , Continuit´ e dans les espaces de Besov. Comm. Partial Differential Equations, 10( 1985), 433-465

    work page 1985

  28. [28]

    Hieber, J

    M. Hieber, J. C. Robinson and Y. Shibata, Mathematical Analysis of the Navier-Stokes Equations, Lec ture Notes in Mathematics , Lecture Notes in Mathematics, vol. 2254. Springer, Berlin , 2020

    work page 2020

  29. [29]

    P. C. Hohenberg, B. I. Halperin, Theory of dynamic critical phenomena , Rev. Mod. Phys. 49(1977), 435-479

    work page 1977

  30. [30]

    Johnsen, Pointwise multiplication of Besov and Triebel-Lizorkin sp aces, Math

    J. Johnsen, Pointwise multiplication of Besov and Triebel-Lizorkin sp aces, Math. Nachr, 175(1995), 85-133

    work page 1995

  31. [31]

    Kalousek, S

    M. Kalousek, S. Mitra, A. Schl¨ omerkemper, Existence of weak solutions to a diffuse interface model invol ving magnetic fluids with unmatched densities , Nonlinear Differ. Equ. Appl. 30(2023), Paper No.52, 53pp

    work page 2023

  32. [32]

    S. G. Kre ˘in, Linear Differential Equations in Banach space , American Mathematical Society, Providence, R.I., Translated from the Russian by J. M.Danskin, Translations o f Mathematical Monographs, Vol. 29, 1971

    work page 1971

  33. [33]

    Lowengrub, L

    J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topologica l transitions , Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 454(1998), 2617-265 4

    work page 1998

  34. [34]

    Meyries, R

    M. Meyries, R. Schnaubelt, Interpolation, embeddings and traces of anisotropic fract ional Sobolev spaces with temporal weights, J. Funct. Anal. 262(2012), 1200-1229

    work page 2012

  35. [35]

    Neˇ cas, I

    J. Neˇ cas, I. Hlavacek, Mathematical theory of elastic and elastico-plastic bodie s: an introduction , Studies in Applied Mechanics 3, Elsevier Scientific Publishing Compan y, Amsterdam-Oxford-New York, 1981

    work page 1981

  36. [36]

    Pazy, Semigroups of linear operators and applications to partial differential equations , Springer, Berlin- Heidelberg-New York, 1983

    A. Pazy, Semigroups of linear operators and applications to partial differential equations , Springer, Berlin- Heidelberg-New York, 1983

    work page 1983

  37. [37]

    Pr¨ uss, G

    J. Pr¨ uss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equ ations, Monographs in Math- ematics, vol. 105. Birkh¨ auser/Springer, [Cham], 2016

    work page 2016

  38. [38]

    M. Reed, B. Simon, Methods of modern mathematical physics: Functional analys is, Elsevier, 2012

    work page 2012

  39. [39]

    J. Shen, X. F. Yang and Q. Wang, Mass and volume conservation in phase field models for binary fluids, Commun. Comput. Phys. 13(2013), 1045-1065

    work page 2013

  40. [40]

    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-incompressi ble form and a linear energy-stable method , Math. Models Methods Appl. Sci. 28(2018), 733-770

    work page 2018

  41. [41]

    de Simon, Unapplicazione della teoria degli integrali singolari all o studio delle equazioni differenziali lineari astratte del primo ordine , Rend

    L. de Simon, Unapplicazione della teoria degli integrali singolari all o studio delle equazioni differenziali lineari astratte del primo ordine , Rend. Sem. Mat. Univ. Padova, 34(1964), 205-223

    work page 1964

  42. [42]

    Simon, Sobolev, Besov and Nikolskii fractional spaces: imbedding s and comparisons for vector valued spaces on an interval , Ann

    J. Simon, Sobolev, Besov and Nikolskii fractional spaces: imbedding s and comparisons for vector valued spaces on an interval , Ann. Mat. Pura Appl. 157(1990), 117-148

    work page 1990

  43. [43]

    Sohr, The Navier-Stokes equations

    H. Sohr, The Navier-Stokes equations. An elementary functional ana lytic approach.[2013 reprint of the 2001 original], Birkh¨ auser/Springer Basel AG, Basel, 2001

    work page 2013

  44. [44]

    Taylor, Partial Differential Equations: III Nonlinear Equations , Applied Mathematical Sciences

    M. Taylor, Partial Differential Equations: III Nonlinear Equations , Applied Mathematical Sciences. Springer, New York, 2010

    work page 2010

  45. [45]

    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(2023), 175-221

    work page 2023

  46. [46]

    Triebel, Interpolation Theory, Function Spaces, Differential Operat ors, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978

    H. Triebel, Interpolation Theory, Function Spaces, Differential Operat ors, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978

    work page 1978

  47. [47]

    Weber, Analysis of Diffuse Interface Models for Two-phase Flows with and Without Surfactants , Ph.D

    J. Weber, Analysis of Diffuse Interface Models for Two-phase Flows with and Without Surfactants , Ph.D. thesis, University Regensburg. https://doi.org/10.5283/epub.3 4247, 2016. QUASI-INCOMPRESSIBLE TWO-PHASE FLOW 29 School of Mathematics and Statistics, Anhui Normal Univers ity, Wuhu 241002, P. R. China Email address : mwfei@ahnu.edu.cn School of Mathemat...

    work page doi:10.5283/epub.3 2016