Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing

Andrea Giorgini; Danica Basari\'c

REVIEW 1 major objections 5 minor 1 cited by

Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing

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Pith's one-line read This paper establishes global-in-time weak solutions for the compressible Navier–Stokes/Cahn–Hilliard system with the singular Flory–Huggins entropy, under γ>3/2, with the phase variable confined to (−1,1) wherever the density is positive.

desk verdict First global weak solutions for compressible NS/Cahn-Hilliard with singular Flory-Huggins potential; the proof is convincing but Lemma 4.1 misstates the pressure integrability exponent. read the letter →

arxiv 2506.07835 v2 pith:ZX7KE5ZS submitted 2025-06-09 math.AP

Danica Basarić , Andrea Giorgini This is my paper · ORCID

classification math.AP MSC 35Q3535D3076N1076T3035K55

keywords compressibleNavier-StokesCahn-HilliardFlory-Hugginsentropysingularpotentialweaksolutionsglobalexistencephaseseparationdensity-dependent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that a compressible mixture of two viscous fluids undergoing phase separation has global-in-time weak solutions even when the free energy contains the singular Flory–Huggins (logarithmic) entropy of mixing, not just the polynomial potentials used in earlier compressible models. The result holds in three-dimensional bounded domains for arbitrarily large finite-energy initial data, provided the density exponent γ satisfies γ>3/2 and the initial relative mass M_r lies strictly between −1 and 1. The authors' new estimates control the chemical potential and the singular entropy directly from the density-dependent Cahn–Hilliard equation, without assuming the density is positive or bounded. The upshot is that physically admissible solutions exist that keep the concentration difference c inside the interval (−1,1) wherever the mixture is present, which is the thermodynamic constraint the logarithmic entropy is meant to enforce.

What carries the argument

The load-bearing mechanism is the control of the density-weighted singular chemical-potential term through a conserved zero-mass identity. Because the relative mass M_r∈(−1,1) is conserved, the quantity ρ_ε(c_ε−M_r) has zero total mass for every approximating solution; choosing c_ε−M_r as a test function in the weak chemical-potential identity (3.8) yields a uniform $L^{2}$_t estimate for ∫ρ_εF'_ε(c_ε)(c_ε−M_r)dx (estimate (4.30)). Since F'_ε is odd, monotone, and behaves like a constant multiple of |F'_ε| outside the band |c_ε|≤K_0, choosing K_0 with |M_r|<K_0<1 converts this single estimate into uniform bounds on ∫ρ_ε|F'_ε(c_ε)|dx, then on μ_ε in $L^{2}$(0,T;$W^{{1,2}}$(Ω)), on √ρ_εF'_ε(c_ε) in $L^{2}$, and on ∇c_ε in $L^{2}$(0,T;$L^{{2p_4}}$) with p_4=3γ/(γ+3)>1. These bounds feed a pressure estimate obtained with a divergence-inverting operator and the renormalized-continuity compactness argument, yielding strong convergence of densities and concentrations; the integral lower-semicontinuity lemma then forces the strict inequality −1<c<1 on {ρ>0}.

What would settle it

Take admissible initial data with M_r=1, equivalently c0=1 on {ρ0>0}; the proof's choice of K_0 with |M_r|<K_0<1 becomes impossible, so the argument converting (4.30) into the $L^{2}$_t bound on ∫ρ_ε|F'_ε(c_ε)|dx fails. If the conclusion of Theorem 1.1 nonetheless held for such data, it would require a mechanism beyond the one constructed; conversely, exhibiting finite-energy initial data satisfying (1.20)–(1.22) for which an approximating sequence fails to keep c in [−1,1], or fails to converge to a weak solution, would refute the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for any time horizon T, any bounded $C^{2}$ domain Ω⊂$ℝ^{3}$, and any adiabatic exponent γ>3/2, the Navier–Stokes/Cahn–Hilliard system (1.1)–(1.10) with free energy f(ρ,c)=ρ^γ/(γ−1)+F(c)−(θ0/2)$c^{2}$ and singular entropy F(c)=(θ/2)[(1+c)ln(1+c)+(1−c)ln(1−c)] admits a weak solution (ρ,u,c) with chemical potential μ whenever the initial data satisfy ρ0≥0, c0∈[−1,1], finite energy E0, total mass M>0, and relative mass M_r=(∫ρ0c0 dx)/M∈(−1,1). The solution obeys the renormalized continuity equation, balance of momentum, transport equation for the concentration, and the chemical-potential identity in weak form, together with the energy inequality. On the set where the density is positive, the phase variable satisfies the strict physical bound −1<c<1 almost everywhere. The proof approximates F by quadratic-growth potentials, derives uniform bounds for the approximating solutions, and passes to the limit with the standard compactness machinery for compressible Navier–Stokes equations; the key new step is a set of estimates for the density-dependent Cahn–Hilliard equation that control ∫ρF'(c) and the full $W^{{1,2}}$-norm of μ using only γ-integrability of ρ with γ>3/2.

Load-bearing premise

The proof's central assumption is that the initial relative mass M_r=(∫ρ0c0 dx)/(∫ρ0 dx) lies strictly between −1 and 1; if only one phase is present initially, so that c0≡±1 wherever ρ0>0, then no constant K_0 with |M_r|<K_0<1 exists and the key estimate controlling the singular chemical-potential term collapses.

Editorial extensions

If this is right

For γ>3/2, the compressible model now has global weak solutions with the physically relevant logarithmic entropy, closing the gap between incompressible and compressible Navier–Stokes/Cahn–Hilliard theory for singular potentials.
The strict bound −1<c<1 on {ρ>0} means solutions respect the interpretation of c as a difference of mass concentrations, so the singular potential can be used without separately enforcing the constraint.
No positivity or boundedness of the initial density is needed, only finite energy and γ-integrability, so the admissible initial-data class is broader than in earlier results requiring ρ*≤ρ0≤ρ*.
The same proof, with minor adjustments, covers two-dimensional domains, more general pressure laws, an additional H(c)lnρ mixing term, and any singular potential F∈C([−1,1])∩C^2(−1,1) with F'(±1)=±∞ and F''≥α>0, as stated in Remark 1.3.
Because the time horizon T is arbitrary, the existence statement is global-in-time and permits arbitrarily large finite-energy initial data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the method's reliance on K_0>|M_r| suggests the argument cannot be pushed to the pure-phase limits M_r→±1; extending existence to initial data with only one phase present would require a mechanism different from the zero-mass estimate (4.30).
Editorial inference: the uniform bound on ∫ρ|F'(c)| may provide quantitative control of phase-separation dissipation, offering a route toward long-time convergence rates to equilibrium that the paper does not address.
Editorial inference: the strategy of controlling a singular nonlinearity through a conserved relative-mass identity may apply to other density-dependent phase-field systems with logarithmic potentials, such as degenerate-mobility or nonlocal Cahn–Hilliard variants, although the paper does not treat those cases.
Editorial inference: a numerical check of the regularized system for γ just above 3/2 and M_r close to 1 would test whether the constant 1/(K_0−|M_r|) in (4.31), which blows up as M_r→1, corresponds to an observable loss of uniform bounds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves global weak solutions for the compressible Navier-Stokes/Cahn-Hilliard system with the singular Flory-Huggins entropy, for gamma > 3/2, and it is the first to treat the singular potential. Previous compressible results (Abels-Feireisl, Elbar-Poulain) only handle polynomial or regular potentials, so the singular entropy was an open problem. The main new ingredient is the uniform estimate on rho_eps F'_eps(c_eps) in L^2(L^1) obtained in Section 4.3, using the zero-mass trick with M_r and the density-dependent Cahn-Hilliard structure. That is genuinely new and does not follow from [4] or [31]. The paper is well-written, and the Lions-Feireisl machinery is applied carefully, with the density convergence and renormalized equation handled in the appendix.

There is one internal inconsistency worth fixing before publication. Lemma 4.1 states p(rho_eps) is precompact in L^q with q(gamma) = min{5/3 - 1/gamma, 3/2}, and the abstract repeats this. But the actual estimate in Section 4.4 gives q(gamma) = 4/3 - 1/(2gamma) for gamma > 6; Remark 4.2 only recovers the stronger exponent for gamma <= 6. For gamma > 6 the claimed exponent is not supported. This is not fatal: the proof only needs q > 1 and uses the derived exponent, so the main theorem survives. However, the convergence statement in Lemma 4.1 and the abstract should be corrected, or the estimate improved. A referee should ask for that.

The hypothesis M_r in (-1,1) is essential for (4.30)-(4.31); the argument breaks down if only one phase is present. This is an explicit hypothesis, not a hidden flaw, and it is physically reasonable. The authors discuss it in Remark 1.2, which is fair. The approximation of the potential follows [31] (second author's own paper) and existence of approximating solutions is taken from [4]—these are legitimate external inputs, not circular.

Overall: the central theorem is new, the proof is credible, and the flaw is local and fixable. I would send this to a serious journal and let a referee check the corrected exponent and the details of Section 4.3. If those check out, it should be accepted. The paper is mainly for researchers working on compressible two-phase flow and the Feireisl-Lions theory; the new estimates will likely be reused. I would cite it and bring it to a reading group.

Referee Report

1 major / 5 minor

Summary. This paper proves the existence of global weak solutions to a three-dimensional compressible Navier–Stokes/Cahn–Hilliard system with the singular Flory–Huggins entropy F(c)=θ/2[(1+c)ln(1+c)+(1−c)ln(1−c)]. The main theorem (Theorem 1.1) requires γ>3/2, finite-energy initial data with c0∈[−1,1], and a relative initial mass M_r=∫ρ0c0/∫ρ0 strictly between −1 and 1. The authors regularize F by the quadratic extensions F_ε of [31], use the existence theory of [4] for the approximate system, and prove uniform estimates for ρ_ε F'_ε(c_ε), μ_ε and c_ε. They then pass to the limit via the Lions–Feireisl strategy, prove strong convergence of densities and concentrations, and conclude that −1<c<1 almost everywhere on {ρ>0}.

Significance. If the exponent statements are corrected, this is a significant advance: it is the first existence result for the compressible phase-field model with the physical logarithmic potential, in the range γ>3/2, and it rigorously enforces the physical bounds on the order parameter. The proof is detailed, follows the standard Lions–Feireisl framework in a careful way, and the paper is transparent about the restriction M_r∈(−1,1) and about the approximation of F. The new chemical-potential and entropy estimates (4.30)–(4.32) are convincing and likely to be useful in later work on singular potentials in compressible phase-field models.

major comments (1)

[§4.4, Lemma 4.1, (1.19)] The claimed pressure integrability q(γ)=min{5/3−1/γ, 3/2} is not established. The estimates in §4.4 yield q(γ)=5/3−1/γ for 3/2<γ<6 and q(γ)=4/3−1/(2γ) for γ>6, and the endpoint γ=6 is not covered by the improved estimate because the Sobolev exponent p=6γ/(7γ−6) equals 1 there, below the range allowed for Bogovskii's operator in Lemma 2.3. Since Lemma 4.1(4.13) and equation (1.19) assert the larger exponent 3/2 for γ>6 (and 1.5 at γ=6), they overclaim the result. Section A.3's use of 'estimate (4.40)' for the L^2-threshold γ≥9/5 is only correct with the improved piecewise exponent. Please correct the statements to the piecewise formula in Remark 4.2 and verify the downstream uses; the existence proof is unaffected because the derived q satisfies q>1 for all γ>3/2.

minor comments (5)

[§4.3(v)] The displayed estimate contains the term θ0ϱεcε^2; based on (3.6) and (1.4) it should be θ0ϱεcε. Please correct the typo.
[§4.4] The test function b(z)=z^ν is not bounded on [0,∞) as stated; please recall the standard truncation (e.g., b_k(z)=min{z^ν,k^ν}) that makes the Bogovskii-based identities legitimate.
[§4.9] After (4.39), the phrase 'letting ψ→1' is informal because ψ∈C_c∞(0,T); the standard approximation by cutoffs should be mentioned.
[§4.3(ii)] In the passage following (4.31), there is a missing variable in 'for any ∈(−1,1)'; it should read 'for any s∈(−1,1)'.
[§1.2 / Remark 1.2] The paper should state explicitly that the case |M_r|=1 is left open; the proof of (4.28)–(4.31) requires a K0 with |M_r|<K0<1, so the current argument does not cover pure-phase initial data.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No empirical free parameters are introduced; constants theta, theta0, gamma are physical inputs. The proof relies on standard tools of the Lions-Feireisl theory (Bogovskii operator, Aubin-Lions lemma, oscillation defect measure) and on structural assumptions on the domain, the viscosity, and the initial data. No new entities are postulated.

assumptions (9)

standard math Existence of Bogovskii operator (Lemma 2.3) on bounded Lipschitz domains
Used in Section 4.4 to construct pressure test functions via the equation div B(f)=f.
standard math Aubin-Lions compactness lemma
Used in Section 4.5 to obtain strong time compactness of the density sequence.
standard math Sobolev embeddings and Poincare inequalities (Lemma 2.1, 2.2, 2.4)
Throughout the proof, essential for the estimates in Sections 4.2 and 4.3.
standard math Feireisl's oscillation defect measure and renormalized continuity equation framework
Used in Appendix A to prove strong convergence of densities for gamma in (3/2, 9/5).
standard math Commutator lemma (from [20]) and Div-Curl lemma (from [19])
Used in Appendix A to derive the effective viscous flux identity (A.5).
domain assumption Domain Omega is bounded and of class C^2
Required for elliptic regularity and boundary conditions in the chemical potential equation.
domain assumption Viscosity coefficients satisfy 0 < eta <= eta(c) <= bar eta, 0 <= lambda(c) <= bar lambda
Assumed in (1.8); ensures Korn-type estimates and positivity of the effective viscosity in (A.5).
domain assumption Initial data satisfy (1.20)-(1.22), including M_r in (-1,1)
Required for the key estimates in Section 4.3; if M_r = +/-1 the proof breaks down.
domain assumption Thermodynamic parameters 0 < theta < theta0
Assumed for the Flory-Huggins potential (1.9); needed for the energy structure.

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Pith. "Pith review of Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing." pith.science (2026). https://pith.science/paper/ZX7KE5ZS

@misc{pith2026250607835,
  author       = {Pith},
  title        = {Pith review of: Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZX7KE5ZS}},
  note         = {Machine review of arXiv:2506.07835}
}

abstract

We study a Navier-Stokes/Cahn-Hilliard system modeling the evolution of a compressible binary mixture of viscous fluids undergoing phase separation. The novelty of this work is a free energy potential including the physically relevant Flory-Huggins (logarithmic) entropy, as opposed to previous studies in the literature, which only consider regular potentials with polynomial growth. Our main result establishes the existence of global-in-time weak solutions in three-dimensional bounded domains for arbitrarily large initial data. The core contribution is the derivation of new estimates for the chemical potential and the Flory-Huggins entropy arising from a density-dependent Cahn-Hilliard equation under minimal assumptions: non-negative $\gamma$-integrable density with $\gamma>\frac32$. In addition, we prove that the phase variable, which represents the difference of the mass concentrations, takes value within the physical interval $(-1,1)$ almost everywhere on the set where the density is positive.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Global strong solutions for 1D compressible Navier-Stokes/Cahn-Hilliard equations with vacuum
math.AP 2026-06 unverdicted novelty 6.0 of 10

Global strong solutions exist and are unique for 1D compressible NS/CH with vacuum without compatibility conditions via singular-in-time weighted estimates.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Abels and E

H. Abels and 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

[2] [31]

Giorgini and R

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

work page 2020

[3] [1]

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

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

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

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

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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´ eaire30, 1175–1190, 2013

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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, 1267–1321, 2024. 28

work page 2024

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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, 1150013, 2012

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Abels, H

H. Abels, H. Garcke, G. Gr¨ un and S. Metzger,Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities, In: Bothe, D., Reusken, A. (eds) Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics. Birkh¨ auser, Cham (2017)

work page 2017