On weak solutions for the stationary Cahn-Hillard-Navier-Stokes equations with singular potential

Dehua Wang; Jiangyu Shuai; Sen Liu; Zhilei Liang

arxiv: 2605.02282 · v1 · submitted 2026-05-04 · 🧮 math.AP

On weak solutions for the stationary Cahn-Hillard-Navier-Stokes equations with singular potential

Zhilei Liang , Sen Liu , Jiangyu Shuai , Dehua Wang This is my paper

Pith reviewed 2026-05-08 18:53 UTC · model grok-4.3

classification 🧮 math.AP

keywords weak solutionsstationary Navier-Stokes-Cahn-Hilliardsingular logarithmic potentialvacuum statescompressible two-phase flowexistence of solutionsadiabatic exponentdiffuse interface

0 comments

The pith

Weak solutions exist for the stationary Navier-Stokes-Cahn-Hilliard system with singular logarithmic free energy and vacuum states.

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

The paper proves existence of weak solutions for the stationary compressible Navier-Stokes-Cahn-Hilliard equations in a three-dimensional bounded domain. The model describes a two-phase fluid mixture with a diffuse interface whose free energy density takes a singular logarithmic form that forces the mass fraction to stay in the physical range while permitting vacuum regions. The stationary setting supplies no evolutionary energy dissipation to control the singularity, and vanishing density creates degeneracy in the equations. The authors restore compactness by regularizing the logarithmic term to remove quadratic growth from anti-diffusion, then obtain uniform estimates via artificial pressure and interpolation before passing to the limit in two stages. The resulting weak solutions satisfy the equations in the distributional sense and obey the physical bounds almost everywhere on the support of the density.

Core claim

We prove the existence of weak solutions in a three-dimensional bounded domain under structural assumptions on the adiabatic exponent. The stationary setting poses two main mathematical challenges: the absence of an energy inequality driven by the evolution process to control the singular potential, and the degeneracy of the density near the vacuum. To address these issues, we introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure and an interpolation argument that controls the desired norm of the density. A two-

What carries the argument

Specialized regularization of the logarithmic term that eliminates quadratic growth from anti-diffusion, together with artificial pressure and interpolation for uniform estimates, followed by a two-level limiting process.

If this is right

Weak solutions exist that satisfy the stationary system in the distributional sense.
The mass fraction remains in the interval between zero and one almost everywhere on the support of the density.
Vacuum states are admissible within the class of weak solutions.
The two-level limiting procedure yields solutions that respect the singular free-energy structure.
The result provides the first existence theory for this steady system that includes both singularity and vacuum.

Where Pith is reading between the lines

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

The regularization technique may extend to related time-dependent compressible mixture models with other singular potentials.
The uniform estimates could inform the design of structure-preserving numerical schemes for phase-separating flows.
Similar compactness arguments might apply to stationary models in unbounded domains once suitable decay conditions are imposed.

Load-bearing premise

The structural assumptions on the adiabatic exponent permit the specialized regularization to cancel quadratic growth from anti-diffusion and thereby restore compactness.

What would settle it

A counterexample or high-resolution numerical computation exhibiting non-existence or violation of the physical bounds for an adiabatic exponent outside the assumed structural class would falsify the result.

Figures

Figures reproduced from arXiv: 2605.02282 by Dehua Wang, Jiangyu Shuai, Sen Liu, Zhilei Liang.

**Figure 1.** Figure 1: The plot of f δ 2 (blue line) and f δ 2 − θc 2 c 2 (red line) in (A), and their derivatives in (B) and (C) With above preparation, we approximate the functions in (1.3)-(1.4) by f δ (ρ, c) := ρ γ−1 + H(c) ln ρ + f δ 2 (c) − θc 2 c 2 , c ∈ (−∞, ∞). (2.5) It is clear that, for fixed δ > 0, the function f δ (ρ, c) is regular with respect to c. Let m1 and m2 be as in (1.5) and |Ω| be the Lebesgue measure of Ω,… view at source ↗

read the original abstract

The stationary Navier--Stokes--Cahn--Hilliard equations are considered, governing the motion of a compressible, two-phase fluid mixture with a diffuse interface. The free energy density in this paper has a singular logarithmic (Flory-uggins) form, ensuring that the mass fraction remains in the physical range and allowing for vacuum states. We prove the existence of weak solutions in a three-dimensional bounded domain under structural assumptions on the adiabatic exponent. The stationary setting poses two main mathematical challenges: the absence of an energy inequality driven by the evolution process to control the singular potential, and the degeneracy of the density near the vacuum. To address these issues, we introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure and an interpolation argument that controls the desired norm of the density. A two-level limiting process then yields a weak solution that satisfies the physical bounds almost everywhere on the support of the density.To our knowledge, this is the first existence result for the steady compressible Navier--Stokes--Cahn--Hilliard system that incorporates both a singular free energy and vacuum regions.

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.

Desk Editor's Note private letter to a colleague

The paper proves existence of weak solutions for the stationary compressible NS-CH system with singular log potential and vacuum, using a tailored regularization plus artificial pressure and two-level limits.

read the letter

The core contribution is an existence result for weak solutions to the stationary Navier-Stokes-Cahn-Hilliard equations in a 3D bounded domain. The free energy uses the singular logarithmic Flory-Huggins form, which enforces the phase variable to stay in [0,1] and permits vacuum regions where density vanishes. This appears to be the first such result for the steady case that combines both features, as the abstract states directly. The stationary setting removes the usual time-integrated energy inequality that would otherwise control the singular term, and the vacuum degeneracy complicates compactness in 3D. The authors address both by introducing a specialized regularization of the log potential that cuts off the quadratic anti-diffusion growth, then adding artificial pressure and applying an interpolation argument to obtain uniform bounds on the density. They pass to the limit in two stages to recover a weak solution that satisfies the physical constraints almost everywhere on the support of the density. The structural assumptions on the adiabatic exponent are the usual ones that guarantee enough integrability for the compressible Navier-Stokes part. The approach looks technically reasonable on the surface and gives credit to prior work on non-stationary or non-singular versions. The potential soft spot is the ordering of the two limits. The regularization is meant to restore compactness, but if the key estimates still depend on the artificial pressure parameter in a way that deteriorates as that parameter vanishes, the first limit could fail to be uniform. The abstract claims the specialized form eliminates the problematic growth, which would fix the issue, but the details of the constants and how they remain controlled need explicit checking in the proofs. This paper is for researchers working on mathematical analysis of diffuse-interface models for compressible multiphase flows. A reader already familiar with artificial pressure methods and singular potentials in stationary systems will see the incremental advance and the specific technical fixes. It is coherent on its own terms and shows honest engagement with the known difficulties, so it deserves peer review even if the limit passage requires more justification in revision.

Referee Report

2 major / 2 minor

Summary. The paper proves existence of weak solutions to the stationary compressible Navier-Stokes-Cahn-Hilliard system in a bounded 3D domain, with singular logarithmic (Flory-Huggins) free energy that permits vacuum states. The strategy combines a specialized regularization of the log term to remove quadratic anti-diffusion growth, an artificial-pressure term, an interpolation argument to obtain uniform density estimates, and a two-level limiting process (regularization parameter followed by artificial pressure) to pass to the limit while preserving physical bounds a.e. on the support of the density. Structural assumptions on the adiabatic exponent are imposed to close the estimates.

Significance. If the limiting arguments can be made rigorous, the result would constitute the first existence theorem for the steady compressible NS-CH system that simultaneously incorporates a singular free energy and vacuum regions. It directly addresses the two principal difficulties of the stationary setting: absence of an evolutionary energy inequality to control the singular potential, and degeneracy of the density near vacuum. The approach of tailored log regularization plus artificial pressure offers a potentially reusable technique for other stationary diffuse-interface models with vacuum.

major comments (2)

[Abstract (two-level limiting process) and the section containing the a-priori estimates] The two-level limiting process (regularization parameter then artificial pressure) is load-bearing for the entire existence claim. The abstract states that uniform estimates are obtained via artificial pressure and interpolation, yet it is not clear from the outline whether the constants in the key interpolation or singular-potential estimates remain independent of the artificial-pressure parameter. If the bound on the regularized logarithmic term deteriorates as the artificial pressure tends to zero, compactness restored by the regularization may be lost before the vacuum degeneracy is controlled, exactly as flagged in the stress-test note.
[Assumptions and the interpolation argument] The structural assumptions on the adiabatic exponent are invoked to close the interpolation argument that controls the desired norm of the density. The precise range (e.g., lower bound on gamma) must be stated explicitly and the dependence of all constants on this exponent tracked through the estimates; otherwise it is impossible to verify that the assumptions are sufficient for the 3D compactness step.

minor comments (2)

[Regularization step] Clarify the precise definition of the specialized regularization of the logarithmic term (e.g., the cutoff or mollification parameter) and verify that it indeed eliminates quadratic growth without introducing new singularities.
[Introduction] Add a short comparison paragraph with existing existence results for compressible NS-CH systems (even those without singular potentials or vacuum) to situate the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help clarify the presentation of the limiting arguments and assumptions. We address each major comment below.

read point-by-point responses

Referee: [Abstract (two-level limiting process) and the section containing the a-priori estimates] The two-level limiting process (regularization parameter then artificial pressure) is load-bearing for the entire existence claim. The abstract states that uniform estimates are obtained via artificial pressure and interpolation, yet it is not clear from the outline whether the constants in the key interpolation or singular-potential estimates remain independent of the artificial-pressure parameter. If the bound on the regularized logarithmic term deteriorates as the artificial pressure tends to zero, compactness restored by the regularization may be lost before the vacuum degeneracy is controlled, exactly as flagged in the stress-test note.

Authors: We appreciate the referee highlighting the need for greater clarity on the uniformity of estimates. The specialized regularization of the logarithmic term is constructed precisely so that the resulting bounds on the singular potential remain independent of the artificial-pressure parameter ε. The interpolation argument is applied after deriving ε-independent controls on the density and velocity fields, ensuring that compactness is retained before passing to the limit ε → 0. The two-level process proceeds by first removing the regularization for fixed ε and then letting ε vanish. We will revise the abstract and the a-priori estimates section to state explicitly that all constants are independent of ε and to detail the order of limits. revision: partial
Referee: [Assumptions and the interpolation argument] The structural assumptions on the adiabatic exponent are invoked to close the interpolation argument that controls the desired norm of the density. The precise range (e.g., lower bound on gamma) must be stated explicitly and the dependence of all constants on this exponent tracked through the estimates; otherwise it is impossible to verify that the assumptions are sufficient for the 3D compactness step.

Authors: We agree that the structural assumptions require more explicit statement. The analysis imposes a lower bound on the adiabatic exponent γ to close the interpolation in three dimensions. We will revise the introduction, the statement of the main theorem, and the estimates section to specify the precise range (γ > 3/2) and to include remarks tracking the dependence of all constants appearing in the estimates on γ. revision: yes

Circularity Check

0 steps flagged

No circularity: self-contained existence proof via a priori estimates and limits

full rationale

The paper derives existence of weak solutions for the stationary Navier-Stokes-Cahn-Hilliard system with singular logarithmic potential through regularization of the free energy, introduction of artificial pressure, interpolation-based uniform estimates, and a two-level limiting process. These steps rely on structural assumptions on the adiabatic exponent and standard PDE techniques to obtain compactness and pass to the limit while preserving physical bounds. No step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the central claim is obtained from the equations and estimates without circular reduction to inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The proof relies on standard functional-analytic tools and a custom regularization; no new physical entities are introduced.

free parameters (1)

adiabatic exponent
Structural assumptions on the exponent are required to obtain uniform estimates and close the compactness argument.

axioms (1)

standard math Standard Sobolev embeddings, interpolation inequalities, and compactness theorems in bounded domains
Invoked to control density norms and pass to the limit after regularization.

pith-pipeline@v0.9.0 · 5520 in / 1122 out tokens · 34977 ms · 2026-05-08T18:53:44.655003+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith.Cost (Jcost = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

F(c) = (θ_0/2)((1+c)ln(1+c) + (1−c)ln(1−c)) − (θ_c/2)c², c ∈ (−1,1).
Foundation/AxiomDischargePlan, Cost/FunctionalEquation n/a — no ratio-symmetric or J-cost structure invoked unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure (ln(1/δ))⁻¹ ρ¹¹ and an interpolation argument.
Constants (parameter-free derivations) n/a — adiabatic exponent is a free structural parameter, not derived unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

γ > 3/2 if ∇×g₁ = 0, γ > 5/3 if ∇×g₁ ≠ 0

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

36 extracted references · 36 canonical work pages

[1]

Abels, On a diffuse interface model for two-phase flows of viscus, incompressible fluids with matched densities, Arch

H. Abels, On a diffuse interface model for two-phase flows of viscus, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194, 463-506, 2009

work page 2009
[2]

Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm

H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289, 45-73, 2009

work page 2009
[3]

Abels, E

H. Abels, E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57, 659-698, 2008

work page 2008
[4]

Abels, H

H. Abels, H. Garcke, G. Grun, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math Mod Meth Appl S., 22, 1150013, 2012

work page 2012
[5]

Abels, J

H. Abels, J. Weber, Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free En- ergies. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhauser, Basel, 2016

work page 2016
[6]

Bresch, C

D. Bresch, C. Burtea, Weak solutions for the stationary anisotropic and nonlocal compressible Navier-Stokes system, J. Math. Pures Appl., 146(9), 183-217, 2021

work page 2021
[7]

Biswas, S

T. Biswas, S. Dharmatti, P. Mahendranath, M. Mohan, On the stationary nonlocal Cahn-Hilliard-Navier- Stokes system: existence, uniqueness and exponential stability, Asymptot. Anal., 125(1-2), 59-99, 2021

work page 2021
[8]

Biswas, S

T. Biswas, S. Dharmatti, M. Mohan, Second order optimality conditions for optimal control problems gov- erned by 2D nonlocal Cahn-Hillard-Navier-Stokes equations, Nonlinear Stud., 28(1), 29-43, 2021 CAHN-HILLIARD-NAVIER-STOKES EQUATIONS WITH SINGULAR POTENTIAL 31

work page 2021
[9]

Basaric, A

D. Basaric, A. Giorgini, Global weak solutions to a compressible Navier-Stokes/Cahn-Hilliard system with singular entropy of mixing, arXiv:2506.07835v1, 2025

work page arXiv 2025
[10]

Colli, S

P. Colli, S. Frigeri, M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hillard-Navier-Stokes system, J. Math. Anal. Appl., 386, 428-444, 2012

work page 2012
[11]

Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, American Mathe- matical Society, Providence, RI, 2010

L. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, American Mathe- matical Society, Providence, RI, 2010

work page 2010
[12]

Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Vol

L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Vol. 74, American Math- ematical Society, 1990

work page 1990
[13]

Feireisl, A

E. Feireisl, A. Novotny, Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhauser, 2009

work page 2009
[14]

Feireisl, A

E. Feireisl, A. Novotny, H. Petzeltova, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 358-392, 2001

work page 2001
[15]

Frehse, M

J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible flow in three dimensions, J. Math. Pures Appl., 97, 85-97, 2012

work page 2012
[16]

Mirnaville, R

A. Mirnaville, R. Temam, On the Cahn-Hilliard-Oono-Navier-Stokes equations with singular potentials, Appl. Anal., 95(12), 2609-2624, 2015

work page 2015
[17]

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

S. Frigeri, On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities, Ann. Inst. Henri Poincare Anal. Non Lin’eaire., 38(3), 647- 687, 2021

work page 2021
[18]

Gilbarg, N

D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn., Grundlehren der Mathematischen Wissenschaften, Vol. 224, SpringerVerlag, 1983

work page 1983
[19]

C. Gal, A. Giorgini, M. Grasselli, A. Poiatti, Global well posedness and convergence to equilibrium for the Abels-Garcke-Grun model with nonlocal free energy, J. Math. Pures Appl., 178, 46-109, 2023

work page 2023
[20]

Giorgini, A

A. Giorgini, A. Miranville, R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51(3), 2535-2574, 2007

work page 2007
[21]

Giorgini, R

A. Giorgini, R. Temam, Weak and strong solutions to the non-homogeneous incompressible Navier-Stokes- Cahn-Hilliard system, J. Math. Pures Appl., 144, 194-249, 2020

work page 2020
[22]

Hohenberg, B

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

work page 1977
[23]

Jiang, C

S. Jiang, C. Zhou, Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations, Ann. Inst. H. Poincare Aal. Non Lineaire., 28, 485-498, 2011

work page 2011
[24]

S. Ko, P. Pustejovska, E. Suli, Finite element approximation of an incompressible chemically reacting non- Newtonian fluid, Math. Mod. Numerical Appl., 52(2), 509-541, 2018

work page 2018
[25]

S. Ko, E. Suli, Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index, Math. Comp., 88(317), 1061-1090, 2019

work page 2019
[26]

Liang, D

Z. Liang, D. Wang, Stationary Cahn-Hilliard-Navier-Stokes equations for the diffuse interface model of com- pressible flows, Math Mod Meth Appl S., 30(12), 2445-2486, 2020

work page 2020
[27]

Liang, D

Z. Liang, D. Wang, Weak solutions to the stationary Cahn-Hilliard/Navier-Stokes equations for compressible fluids, J. Nonlinear Sci., 32, 41, 2022

work page 2022
[28]

Lions, Mathematical topics in fluid mechanics, Vol

P. Lions, Mathematical topics in fluid mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, Vol. 10, Oxford Science Publications/The Clarendon Press, 1998

work page 1998
[29]

Mucha, M

P. Mucha, M. Pokorny, On a new approach to the issue of existence and regularity for the steady compressible Navier-Stokes equations, Nonlinearity, 19, 1747-1768, 2006

work page 2006
[30]

Mucha, M

P. Mucha, M. Pokorny, E. Zatorska, Existence of stationary weak solutions for compressible heat conducting flows, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, 2595-2662, 2018

work page 2018
[31]

Novotny, I

A. Novotny, I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, Vol. 27, Oxford Univ, Press, 2004

work page 2004
[32]

C. Hurm, P. Knopf, A. Poiatti, Nonlocal-to-local convergence rates for strong solutions to a Navier-Stokes- CahnHilliard system with singular potential, Comm. Partial Differential Equations., 49(2), 1-40, 2024

work page 2024
[33]

Lowengrub, L

J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. Ser. A., 454, 2617-2654, 1998

work page 1998
[34]

Plotnikov, W

P. Plotnikov, W. Weigant, Steady 3D viscous compressible flows with adiabatic exponentγ∈(1,∞), J. Math. Pures Appl., 104, 58-82, 2015

work page 2015
[35]

Stein, Singular integrals and differentiability properties of functions, Bulletin of the London Mathematical Society 5, 1973

E. Stein, Singular integrals and differentiability properties of functions, Bulletin of the London Mathematical Society 5, 1973

work page 1973
[36]

Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, 1989 32 Z

W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, 1989 32 Z. LIANG, S. LIU, J. SHUAI, AND D. WANG School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China. Email address:liangzl@swufe.edu.cn School of Mathematics, Southwestern University of Finance and Economics, ...

work page 1989

[1] [1]

Abels, On a diffuse interface model for two-phase flows of viscus, incompressible fluids with matched densities, Arch

H. Abels, On a diffuse interface model for two-phase flows of viscus, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194, 463-506, 2009

work page 2009

[2] [2]

Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm

H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289, 45-73, 2009

work page 2009

[3] [3]

Abels, E

H. Abels, E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57, 659-698, 2008

work page 2008

[4] [4]

Abels, H

H. Abels, H. Garcke, G. Grun, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math Mod Meth Appl S., 22, 1150013, 2012

work page 2012

[5] [5]

Abels, J

H. Abels, J. Weber, Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free En- ergies. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhauser, Basel, 2016

work page 2016

[6] [6]

Bresch, C

D. Bresch, C. Burtea, Weak solutions for the stationary anisotropic and nonlocal compressible Navier-Stokes system, J. Math. Pures Appl., 146(9), 183-217, 2021

work page 2021

[7] [7]

Biswas, S

T. Biswas, S. Dharmatti, P. Mahendranath, M. Mohan, On the stationary nonlocal Cahn-Hilliard-Navier- Stokes system: existence, uniqueness and exponential stability, Asymptot. Anal., 125(1-2), 59-99, 2021

work page 2021

[8] [8]

Biswas, S

T. Biswas, S. Dharmatti, M. Mohan, Second order optimality conditions for optimal control problems gov- erned by 2D nonlocal Cahn-Hillard-Navier-Stokes equations, Nonlinear Stud., 28(1), 29-43, 2021 CAHN-HILLIARD-NAVIER-STOKES EQUATIONS WITH SINGULAR POTENTIAL 31

work page 2021

[9] [9]

Basaric, A

D. Basaric, A. Giorgini, Global weak solutions to a compressible Navier-Stokes/Cahn-Hilliard system with singular entropy of mixing, arXiv:2506.07835v1, 2025

work page arXiv 2025

[10] [10]

Colli, S

P. Colli, S. Frigeri, M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hillard-Navier-Stokes system, J. Math. Anal. Appl., 386, 428-444, 2012

work page 2012

[11] [11]

Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, American Mathe- matical Society, Providence, RI, 2010

L. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, American Mathe- matical Society, Providence, RI, 2010

work page 2010

[12] [12]

Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Vol

L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Vol. 74, American Math- ematical Society, 1990

work page 1990

[13] [13]

Feireisl, A

E. Feireisl, A. Novotny, Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhauser, 2009

work page 2009

[14] [14]

Feireisl, A

E. Feireisl, A. Novotny, H. Petzeltova, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 358-392, 2001

work page 2001

[15] [15]

Frehse, M

J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible flow in three dimensions, J. Math. Pures Appl., 97, 85-97, 2012

work page 2012

[16] [16]

Mirnaville, R

A. Mirnaville, R. Temam, On the Cahn-Hilliard-Oono-Navier-Stokes equations with singular potentials, Appl. Anal., 95(12), 2609-2624, 2015

work page 2015

[17] [17]

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

S. Frigeri, On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities, Ann. Inst. Henri Poincare Anal. Non Lin’eaire., 38(3), 647- 687, 2021

work page 2021

[18] [18]

Gilbarg, N

D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn., Grundlehren der Mathematischen Wissenschaften, Vol. 224, SpringerVerlag, 1983

work page 1983

[19] [19]

C. Gal, A. Giorgini, M. Grasselli, A. Poiatti, Global well posedness and convergence to equilibrium for the Abels-Garcke-Grun model with nonlocal free energy, J. Math. Pures Appl., 178, 46-109, 2023

work page 2023

[20] [20]

Giorgini, A

A. Giorgini, A. Miranville, R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51(3), 2535-2574, 2007

work page 2007

[21] [21]

Giorgini, R

A. Giorgini, R. Temam, Weak and strong solutions to the non-homogeneous incompressible Navier-Stokes- Cahn-Hilliard system, J. Math. Pures Appl., 144, 194-249, 2020

work page 2020

[22] [22]

Hohenberg, B

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

work page 1977

[23] [23]

Jiang, C

S. Jiang, C. Zhou, Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations, Ann. Inst. H. Poincare Aal. Non Lineaire., 28, 485-498, 2011

work page 2011

[24] [24]

S. Ko, P. Pustejovska, E. Suli, Finite element approximation of an incompressible chemically reacting non- Newtonian fluid, Math. Mod. Numerical Appl., 52(2), 509-541, 2018

work page 2018

[25] [25]

S. Ko, E. Suli, Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index, Math. Comp., 88(317), 1061-1090, 2019

work page 2019

[26] [26]

Liang, D

Z. Liang, D. Wang, Stationary Cahn-Hilliard-Navier-Stokes equations for the diffuse interface model of com- pressible flows, Math Mod Meth Appl S., 30(12), 2445-2486, 2020

work page 2020

[27] [27]

Liang, D

Z. Liang, D. Wang, Weak solutions to the stationary Cahn-Hilliard/Navier-Stokes equations for compressible fluids, J. Nonlinear Sci., 32, 41, 2022

work page 2022

[28] [28]

Lions, Mathematical topics in fluid mechanics, Vol

P. Lions, Mathematical topics in fluid mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, Vol. 10, Oxford Science Publications/The Clarendon Press, 1998

work page 1998

[29] [29]

Mucha, M

P. Mucha, M. Pokorny, On a new approach to the issue of existence and regularity for the steady compressible Navier-Stokes equations, Nonlinearity, 19, 1747-1768, 2006

work page 2006

[30] [30]

Mucha, M

P. Mucha, M. Pokorny, E. Zatorska, Existence of stationary weak solutions for compressible heat conducting flows, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, 2595-2662, 2018

work page 2018

[31] [31]

Novotny, I

A. Novotny, I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, Vol. 27, Oxford Univ, Press, 2004

work page 2004

[32] [32]

C. Hurm, P. Knopf, A. Poiatti, Nonlocal-to-local convergence rates for strong solutions to a Navier-Stokes- CahnHilliard system with singular potential, Comm. Partial Differential Equations., 49(2), 1-40, 2024

work page 2024

[33] [33]

Lowengrub, L

J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. Ser. A., 454, 2617-2654, 1998

work page 1998

[34] [34]

Plotnikov, W

P. Plotnikov, W. Weigant, Steady 3D viscous compressible flows with adiabatic exponentγ∈(1,∞), J. Math. Pures Appl., 104, 58-82, 2015

work page 2015

[35] [35]

Stein, Singular integrals and differentiability properties of functions, Bulletin of the London Mathematical Society 5, 1973

E. Stein, Singular integrals and differentiability properties of functions, Bulletin of the London Mathematical Society 5, 1973

work page 1973

[36] [36]

Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, 1989 32 Z

W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, 1989 32 Z. LIANG, S. LIU, J. SHUAI, AND D. WANG School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China. Email address:liangzl@swufe.edu.cn School of Mathematics, Southwestern University of Finance and Economics, ...

work page 1989