Bounded solutions and their asymptotics for a doubly nonlinear Cahn-Hilliard system

Elena Bonetti; Giuseppe Tomassetti; Luca Scarpa; Pierluigi Colli

REVIEW 2 major objections 4 minor 32 references

Bounded solutions and their asymptotics for a doubly nonlinear Cahn-Hilliard system

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a doubly nonlinear Cahn–Hilliard system has a unique bounded solution even when the bulk potential is singular and the viscosity nonlinearity is arbitrary, and that both regularization limits converge.

desk verdict The existence theorem is a real advance and the maximum-principle argument is the genuine novelty; the ε→0 limit has one explicitly formal, load-bearing estimate in Section 4.3 that should be fixed before that theorem is relied on. read the letter →

arxiv 1908.02079 v1 pith:64JRHK46 submitted 2019-08-06 math.AP

Elena Bonetti , Pierluigi Colli , Luca Scarpa , Giuseppe Tomassetti This is my paper · ORCID

classification math.AP MSC 35B2535D3535G3135K5274N2074N25

keywords Cahn-Hilliardequationnonlinearviscositydoublysystemsingularpotentialsboundedsolutionsmaximumprincipleasymptoticlimitsregularization

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

The paper studies a Cahn–Hilliard type model of phase separation in which the chemical potential contains two extra dissipation terms: a viscosity $\varepsilon \partial_t u$ and an interface-energy term $-\delta\Delta u$, plus a possibly multivalued nonlinearity $\beta(\partial_t u)$ and a bulk potential $\psi(u)$ that may be singular at the edges of its domain. Its first result is that, for fixed positive $\varepsilon,\delta$, this doubly nonlinear system has a unique solution with the concentration $u$ staying strictly inside the domain of $\psi$, without any growth assumption on $\beta$ or $\psi$; the proof rests on a maximum-principle argument performed on time discretizations. Its second and third results show what happens when each regularization is removed: as $\varepsilon\to 0$ with $\delta$ fixed, solutions converge to a limit problem that can accommodate noncoercive multivalued graphs such as a sign function, and as $\delta\to 0$ with $\varepsilon$ fixed, the whole family converges with an explicit rate. A sympathetic reader should care because singular logarithmic potentials and multivalued viscosity laws are the physically relevant choices, and earlier well-posedness results needed polynomial growth conditions.

What carries the argument

The argument is carried by a time-discretization maximum principle. The regularized system is rewritten as a doubly nonlinear inclusion of the form $\partial\Xi((v^k-v^{k-1})/\tau)+A(v^k)\ni f^k$, and each discrete step is tested against the positive part $(v^k-b'_0)_+$ and $(a'_0-v^k)_+$. The sign of the potential term is controlled by the fast growth of $\psi'$ near the edges of $(a,b)$ and by properties of the resolvent $(I+\lambda\gamma)^{-1}$, forcing every discrete value to remain between $a'_0$ and $b'_0$; passing to the limit in the discretization and in the standard smoothings of the monotone graphs gives the uniform bound on $u$ that makes the singular nonlinearities well defined. The same boundedness is then used as an a priori estimate to let $\varepsilon$ or $\delta$ tend to zero, with compactness results and monotone graph closure arguments identifying the limit selection $\xi\in\beta(\partial_t u)$.

What would settle it

Carry out the omitted smoothing-regularization computation behind Section 4.3. If the calculation produces boundary terms on $\partial\Omega$ or terms involving $\beta'_\lambda(\partial_t u_\lambda)$ that cannot be bounded independently of the smoothing parameter $\lambda$, then the bound (4.5)–(4.6) is not available and Theorem 2.4's $L^\infty$ conclusion would need weaker alternatives. A concrete numerical test of the same phenomenon: take $\beta$ as the sign graph and $\psi$ with $|\psi''(r)|\le M(1+|r|^5)$, and check whether $\sup_{\varepsilon>0}\|u_\varepsilon\|_{L^\infty(Q)}$ is finite as $\varepsilon\to 0$ for an initial datum obeying (2.20)–(2.22).

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes three statements. First, under the hypotheses (2.2)–(2.10), Theorem 2.2 gives a unique triplet $(u,\mu,\xi)$ with $u\in W^{1,\infty}(0,T;H)\cap H^1(0,T;V)\cap L^\infty(0,T;W_n)$, with $a'_0\le u\le b'_0$ almost everywhere for a compact subinterval $[a'_0,b'_0]\subset(a,b)$, with $\xi\in\beta(\partial_t u)$ almost everywhere, and with (2.16)–(2.18) holding. This removes the growth restrictions on $\beta$ and $\psi$ that earlier existence proofs required. Second, Theorem 2.4 says that with $\delta>0$ fixed and data satisfying (2.20)–(2.23) or (2.36), the family of solutions computed with $\varepsilon>0$ has a subsequence converging to a solution of the $\varepsilon=0$ limit problem, including the selection $\xi\in\beta(\partial_t u)$ in the limit. Third, Theorem 2.6 says that with $\varepsilon>0$ fixed and data satisfying (2.37)–(2.42), the solutions converge as $\delta\downarrow 0$ to the $\delta=0$ limit problem, and the difference obeys the rate estimate (2.50): $\|\mu_\delta-\mu\|_{L^2(0,T;V_0)}+\|u_\delta-u\|_{H^1(0,T;H)}\le M(\delta^{1/4}+\|u_{0\delta}-u_0\|_H+\|g_\delta-g\|_{L^2(0,T;H)})$.

Load-bearing premise

The load-bearing step is the third a priori estimate in Section 4.3, which the authors derive by formally testing the chemical-potential inclusion against $-\delta\Delta u_t+\partial_t\gamma(u)$ and describe as formal, deferring the rigorous justification to a smoothing regularization of $\beta$ that is only sketched in words; if that estimate cannot be made rigorous, the uniform $L^\infty$ bounds on $u$ and $\xi$ that the $\varepsilon\to 0$ theorem needs would collapse.

Editorial extensions

If this is right

For fixed $\varepsilon,\delta>0$, the system is well posed for physically natural singular potentials such as the logarithmic double-well potential, with the solution confined to a compact subinterval of the potential's domain.
With $\delta$ fixed, sending $\varepsilon$ to zero shows that the diffusive term alone can handle a noncoercive, multivalued viscosity law, so the vanishing-viscosity limit has a meaningful solution.
With $\varepsilon$ fixed, sending $\delta$ to zero converges the whole family to the hysteresis-type limit equation, and the error is controlled by $\delta^{1/4}$ plus the approximation errors in the initial data and forcing term.
In both limits, $\psi'(u)$ and the selection $\xi$ stay in bounded spaces, so the nonlinearities do not leave the framework even though they were singular or multivalued.

Reading between the lines

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

The same positive-part test on time-discretization steps should extend to other doubly nonlinear gradient-flow systems with two competing regularizations, for instance with nonconstant mobility or different boundary conditions, because the sign argument only uses monotonicity and the steepness of $\psi$ outside an interval.
The rate in (2.50) likely has room to move: the proof yields $\delta^{1/4}$ from balancing $\sqrt\delta$ and $\delta^{3/4}$ estimates, and a sharper comparison of the limit problem might show whether $\delta^{1/2}$ or a similar improved rate is attainable.
If the formal Section 4.3 estimate were to fail, the alternative assumption (2.36) in Theorem 2.4 suggests a natural fallback: retain well-posedness but replace the $L^\infty$ regularity assertions with the corresponding $L^2$ statements rather than abandoning the $\varepsilon\to0$ limit.
The boundedness argument might be adapted to prove existence for logarithmic potentials in related phase-separation models with hysteresis, since the only point where the potential enters is via $\psi'$ exceeding a fixed threshold outside the interval $[a'_0,b'_0]$.

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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

Here is what I would tell you before you decide whether to spend time on arXiv:1908.02079. The paper's real contribution is Theorem 2.2: existence and uniqueness of bounded solutions for the doubly nonlinear Cahn–Hilliard system with no growth assumptions on the maximal monotone graph β and no growth assumptions on the singular potential ψ. That genuinely improves the authors' earlier paper [5] and the Miranville–Schimperna and Miranville–Zelik results. The engine is a nonstandard maximum-principle argument on the time-discretized problem, and the upper-bound half is written in enough detail to follow. The lower half is dismissed with "we omit the details," but it is the mirror image of the upper bound, so I do not count that as a real flaw.

The two asymptotic theorems are the secondary content. The δ→0 limit in Theorem 2.6 is, to my reading, the cleaner half: the estimates are uniform, whole-sequence convergence is obtained from a Gronwall comparison of two solutions, and the rate (2.50) is explicit. That part holds up.

The soft spot is in the ε→0 limit, Theorem 2.4. Section 4.3 contains the estimate that produces the essential uniform bounds (4.5)–(4.6), and the authors themselves mark it as formal. The test function −δΔ∂tuε + ∂tγ(uε) is not legitimate at the regularity level established up to that point: only ∂tuε ∈ L2(0,T;V) is known, so −δΔ∂tuε is a distribution, ξε is not differentiable, and δ∫∇ξε·∇∂tuε has no meaning for an unregularized maximal monotone β. The paper says a Yosida regularization of β would make it rigorous and that the constants would be independent of λ, but the computation is not shown. Since (4.5)–(4.6) are what give the L∞ bounds on uε and ξε used in Section 4.5, the ε→0 part of Theorem 2.4 is not yet fully established. This is a fillable gap rather than a contradiction, but it is load-bearing. The rest of the ε→0 passage is standard compactness plus maximal monotonicity, and the initial-data Lemma 4.1 is handled carefully.

The heavy use of [5] for technical estimates did not bother me. It is their own published work, and the new theorem is precisely the removal of growth conditions; citing the machinery is not circular.

My bottom line: the existence theorem is a real advance and deserves referee time. The δ→0 half is in good shape. Before I would rely on Theorem 2.4, I would want the §4.3 estimate either proved or replaced by a reference where it is proved. Send it to a serious referee, with a note asking the referee to check that estimate.

Referee Report

2 major / 4 minor

Summary. The paper studies the doubly nonlinear Cahn–Hilliard system (1.1)–(1.4), in which the chemical potential contains both a viscous term ε∂tu and a diffusive term −δΔu, with a maximal monotone nonlinearity β acting on ∂tu and a possibly singular potential ψ. The main results are: Theorem 2.2, existence and uniqueness of a bounded solution with values in a compact subinterval of the effective domain of ψ, without polynomial growth restrictions on β or ψ; Theorem 2.3, continuous dependence; Theorem 2.4, convergence as ε → 0 with δ > 0 fixed; and Theorem 2.6, convergence as δ → 0 with ε > 0 fixed, including the rate estimate (2.50). The existence proof combines a Yosida-type approximation with a nonstandard maximum-principle argument for time-discretized problems, and the asymptotic proofs use energy estimates, compactness, and monotonicity. The paper explicitly labels the third estimate of Section 4.3 as formal and defers its rigorous justification to a Yosida regularization that is described only in words.

Significance. If the formal estimates are supplied, the results are a genuine advance: the no-growth existence and uniqueness theorem extends previous work in [5] to singular ψ and β, and the two asymptotic limits answer natural regularization questions for this system. The maximum-principle argument for time-discretizations is nonstandard and is the main technical novelty of the existence proof. The δ → 0 rate estimate (2.50) is a concrete quantitative result that goes beyond mere compactness. The paper is also transparent about the formal step, which makes the gap identifiable rather than hidden. However, the proof of Theorem 2.4 is incomplete as written because the uniform bounds (4.5)–(4.6) rest on an unjustified formal computation; a parallel gap appears in the δ → 0 proof in Section 5.3. These gaps are fillable, and no internal inconsistency or circularity is apparent.

major comments (2)

[Section 4.3, Eqs. (4.5)–(4.6)] The third a priori estimate, which yields the uniform bounds (4.5)–(4.6), is explicitly formal. At this point the available regularity is ∂tuε ∈ L2(0,T;V) and ξε ∈ L∞(0,T;H), so the test function −δΔ∂tuε + ∂tγ(uε) is not legitimate: −Δ∂tuε is only a distribution, and the displayed identity contains δ∫∇ξε·∇∂tuε and ∫γ′(uε)ξε∂tuε, neither of which is meaningful at this regularity. The authors state that a rigorous version can be obtained by replacing β with its Yosida approximation and that the resulting estimate is independent of λ, but no proof is supplied. These bounds are load-bearing: Section 4.5 uses (4.5)–(4.6) to pass to the limit and to identify ξ ∈ β(∂tu). Theorem 2.4 is therefore not fully proved as written. This is a fillable gap, but it must be closed by a complete regularization argument with constants independent of both ε and the regularization parameter.
[Section 5.3, Eqs. (5.15)–(5.16)] The estimate leading to (5.15)–(5.16) is obtained by 'formally testing' (2.17) by −δ1/2Δ∂tuδ. At this stage ξδ is only known to lie in L∞(0,T;H), so the term δ1/2∫∇ξδ·∇∂tuδ is not a well-defined quantity, and no Yosida regularization or time-discretization argument is given for this computation. Since (5.15) and (5.16) are used in Section 5.4 to obtain compactness and the rate estimate (2.50), the proof of Theorem 2.6 has the same type of gap. If the regularization argument for Section 4.3 is carried out, the same argument should be adapted and written out for Section 5.3.

minor comments (4)

[Theorem 2.6, Eq. (2.48)] Equation (2.48) contains a sign error: the limit equation should contain +g(t), not −g(t), consistent with (2.17) and with the computations in Sections 5.1 and 5.3. Please verify and correct this typo.
[Section 5.4, convergence list] In the convergence list before (5.17), the line 'δ1/2u → 0 in H1(0,T;V)' should read 'δ1/2uδ → 0'; the subscript is missing.
[Section 3.1, Proposition 3.1 application] The functional Φ is defined as Φ(v) := 1/2∫(|∇v|² + λ|v2|); this should be λ|v|².
[Section 3.1, maximum principle] The lower-bound half of the maximum-principle argument is omitted with the note 'for brevity'. Since the upper-bound argument uses specific sign choices for b′0, the lower-bound analogue should be written out or the sign conventions should be explained, for completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the load-bearing estimates are derived from the PDE and independent compactness/monotonicity arguments, and the self-citations are technical borrowings rather than assumed conclusions.

full rationale

Walking the claimed derivation chain: Theorem 2.2 is proved by a Yosida/truncation approximation in Section 3.1. Existence of the smooth approximating pair (u_lambda, mu_lambda) is imported from the authors' earlier paper [5], but [5] is a published, parameter-free existence theorem for the same regularized system under growth assumptions, and it does not contain the target no-growth boundedness result; the new maximum-principle argument using Proposition 3.1 and the Rothe discretization supplies the L-infinity control independently. The continuous-dependence estimate in Theorem 2.3 is obtained by a direct Gronwall estimate on differences, not by assuming the result. For Theorem 2.4, the epsilon-to-0 limit is based on the a priori estimates (4.1)-(4.6) derived from testing the equations, the Young inequality, the compactness inequality (2.1), and Lemma 4.1; the identification xi in beta(u_t) is a standard monotonicity and weak-strong closure argument. The delta-to-0 limit and the rate estimate (2.50) in Theorem 2.6 are likewise obtained from direct estimates, the maximum principle with constants depending only on ||mu_delta - g_delta||_{L^infty(Q)}, and the difference estimate (5.17). The only flagged weakness is that the third estimate in Section 4.3 is explicitly called formal and deferred to a Yosida regularization described only in words; this is a rigor gap that, if unfillable, would threaten Theorem 2.4, but it is not a circularity because the estimate is not assumed as the conclusion and the authors do not invoke the target theorem to justify it. There are no fitted inputs, no definitional coincidences, and no self-citation chain that forces the stated theorems. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper contains no fitted numbers, no data-driven constants, and no newly postulated physical entities. Its results rest on standard analytic tools and the explicitly stated hypotheses on the potential and the dissipation function.

assumptions (7)

standard math Proposition 3.1 (from Roubicek [28, Prop. 11.6]): existence and uniform estimates for time-discretized doubly nonlinear inclusions.
Invoked in Section 3.1 to produce Rothe approximations of the Yosida-regularized problem and the convergence in (3.9).
standard math Lemma 2.1 (variation of Lions compactness lemma [19, Lem. 5.1]): interpolation inequality (2.1).
Used in Section 4.2 to absorb the L2 norm of the time derivative into the gradient term plus a compact error.
standard math Aubin-Lions-Simon compactness lemma (Simon [31, Cor. 4]).
Used in Sections 4.5 and 5.4 to obtain strong time-compactness of the solution sequences.
standard math Convex analysis and maximal monotone operator theory, including Yosida approximations and the Young inequality for convex conjugates (Barbu [3]).
Applied throughout for subdifferentials, regularized inclusions, and monotonicity arguments.
standard math Elliptic regularity theory on smooth bounded domains.
Used in Section 3.1 and Theorem 2.4 to obtain H3 regularity of the chemical potential.
domain assumption Assumptions (2.2)-(2.8): psi in C2(a,b), psi >= 0, psi' blows up at the endpoints of (a,b), psi'' > -K, and beta = d beta-hat for a proper convex lower semicontinuous beta-hat with beta-hat(0) = 0.
These are the hypotheses defining the class of potentials and dissipation functions for which the existence theorem is stated.
domain assumption The boundary value problem (1.1)-(1.4) with normal derivative of u equal to zero and mu equal to zero in a smooth bounded domain Omega in R3.
The analysis is specifically for this system; conclusions are conditional on the model and boundary conditions.

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Pith. "Pith review of Bounded solutions and their asymptotics for a doubly nonlinear Cahn-Hilliard system." pith.science (2026). https://pith.science/paper/64JRHK46

@misc{pith2026190802079,
  author       = {Pith},
  title        = {Pith review of: Bounded solutions and their asymptotics for a doubly nonlinear Cahn-Hilliard system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64JRHK46}},
  note         = {Machine review of arXiv:1908.02079}
}

read the original abstract

In this paper we deal with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscosity and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard maximum-principle argument for time-discretizations of doubly nonlinear equations. Possibly including singular potentials, this novel result brings improvements over previous approaches to this problem. Secondly, under suitable assumptions on the data, we show the convergence of solutions to the respective limit problems once either of the two regularization parameters vanishes.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Bonetti, P

E. Bonetti, P. Colli, L. Scarpa and G. Tomassetti. A doubl y nonlinear Cahn-Hilliard system with nonlinear viscosity. Commun. Pure Appl. Anal. 17 (2018), 1001–1022

work page 2018

[2] [1]

Agosti, P.F

A. Agosti, P.F. Antonietti, P. Ciarletta, M. Grasselli, and M. Verani. A Cahn-Hilliard- type equation with application to tumor growth dynamics. Math. Methods Appl. Sci. 40 (2017), 7598–7626

work page 2017

[3] [2]

Bagagiolo and A

F. Bagagiolo and A. Visintin. Hysteresis in ﬁltration th rough porous media. Z. Anal. Anwendungen 19 (2000), 977–997

work page 2000

[4] [3]

Nonlinear diﬀerential equations of monotone types in Banach spaces

V. Barbu. “Nonlinear diﬀerential equations of monotone types in Banach spaces”. Springer Monographs in Mathematics. Springer, New York (2010). Asymptotics for a doubly nonlinear CH system 25

work page 2010

[5] [4]

Bonetti, P

E. Bonetti, P. Colli and G. Tomassetti. A non-smooth regu larization of a forward- backward parabolic equation. Math. Models Methods Appl. Sci. 27 (2017), 641–661

work page 2017

[6] [6]

Bertozzi, S.Esedoglu, and A

A.L. Bertozzi, S.Esedoglu, and A. Gillette. Inpainting of binary images using the Cahn–Hilliard equation. IEEE Trans. Image Process. 16 (2007), 285–291

work page 2007

[7] [7]

Botkin, M

N.D. Botkin, M. Brokate and E. El Behi-Gornostaeva. One- phase ﬂow in porous media with hysteresis. Physica B 486 (2016), 183–186

work page 2016

[8] [8]

J.W. Cahn. On spinodal decomposition. Acta Metall. 9 (1961), 795–801

work page 1961