{"id":"7229eaf3-9cc1-4048-8a03-995118e172d9","arxiv_id":"1908.02079","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A doubly nonlinear Cahn-Hilliard system is shown to admit unique bounded solutions without growth assumptions on the singular potentials, and the solutions converge to limit problems as the viscosity or diffusion regularization vanishes.","lead":"This maths paper proves that a doubly nonlinear Cahn-Hilliard phase-separation model has a unique bounded solution even when the nonlinear potentials are singular, and it shows what happens as two smoothing parameters go to zero. The results give rigorous foundations for widely used phase-field models in materials science, including logarithmic potentials that previously required extra growth conditions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ε→0 limit in Theorem 2.4 depends on the formal §4.3 estimate; the Yosida regularization that would justify it is not carried out, so the L∞ bounds (4.5)–(4.6) are not yet fully established.","rationale":"After reading the full manuscript, I find the reader's assessment accurate. The main existence argument in Section 3 is quite detailed: Proposition 3.1 supplies a Rothe sequence, the maximum-principle test against (u^k−b'_0)_+ is written out, and the estimate (3.19)–(3.20) follows by testing with −Δ∂tuλ. The lower bound is only said to be similar, but it is symmetric and not a serious obstacle. Section 5 (δ→0) is also reasonably complete, with explicit Cauchy estimates and rate (2.50). The weakest point is indeed Section 4.3. The formal test function involves second-order time-space derivatives that are not available from the established regularity, and the term δ∫∇ξ·∇∂tu is meaningless for multivalued β without regularization. The text explicitly acknowledges this and describes a Yosida regularization without performing it. Because (4.5)–(4.6) are used in the passage to the limit for Theorem 2.4, the ε→0 theorem is conditional on that regularization being carried out. I do not see an internal contradiction; the concern is a gap in rigor, not a false statement. Hence the conditional verdict should stand unchanged.","tokens_in":22868,"tokens_out":16736,"duration_ms":175059,"concrete_test":"Write out the Yosida-regularized analogue of Section 4.3: replace β by βλ in (2.17), prove the displayed energy identity rigorously for the λ-regularized problem (by time discretization or elliptic regularity), and show all constants in (4.5)–(4.6) are independent of both λ and ε. Then pass λ→0 before ε→0, checking convergence of the nonnegative terms β'_λ(∂tuλ)|∇∂tuλ|² and ∫γ'(uλ)ξλ∂tuλ to the required limits. If the λ-uniform estimate holds, the formal step is valid; if constants blow up as λ→0 or a boundary term in the integration by parts cannot be controlled, Theorem 2.4's ε→0 limit needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.3 derives the uniform bounds (4.5)–(4.6) by testing the chemical-potential identity (2.17) against −δΔ∂tuε + ∂tγ(uε). At that point only ∂tuε ∈ L2(0,T;V) is known, so −δΔ∂tuε is a distribution in L2(0,T;V*), and ξε ∈ L∞(0,T;H) is not differentiable; the displayed identity also contains δ∫∇ξε·∇∂tuε, which has no meaning for the unregularized maximal monotone β. The authors state that the estimate is formal and defer a rigorous version to a Yosida regularization of β that is described only in words. These bounds are load-bearing: (4.5)–(4.6) give uε ∈ L∞(0,T;Wn), γ(uε) ∈ L∞(0,T;H), and ξε ∈ L∞(0,T;H), and they are used in Section 4.5 to pass to the limit and identify ξ ∈ β(∂tu). If the Yosida regularization cannot be executed with constants independent of ε and λ, the ε→0 part of Theorem 2.4 is not proved. This is a fillable gap rather than an observed contradiction, but it is exactly the kind of step that should be verified before relying on the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23075,"tokens_out":23665,"duration_ms":218249,"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":[{"comment":"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":"Section 4.3, Eqs. (4.5)–(4.6)"},{"comment":"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.","section":"Section 5.3, Eqs. (5.15)–(5.16)"}],"minor_comments":[{"comment":"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":"Theorem 2.6, Eq. (2.48)"},{"comment":"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":"Section 5.4, convergence list"},{"comment":"The functional Φ is defined as Φ(v) := 1/2∫(|∇v|² + λ|v2|); this should be λ|v|².","section":"Section 3.1, Proposition 3.1 application"},{"comment":"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.","section":"Section 3.1, maximum principle"}],"recommendation":"major_revision","confidential_remarks":"The substantive obstacle is the pair of formal estimates in Sections 4.3 and 5.3; I believe they can be repaired by a Yosida regularization with explicit λ-independent estimates. The sign typo in (2.48) should be corrected. The paper is within scope and otherwise carefully written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nMy 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.","headline":"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.","tokens_in":23675,"tokens_out":2605,"would_cite":true,"duration_ms":26540,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B25","35D35","35G31","35K52","74N20","74N25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Cahn-Hilliard equation","nonlinear viscosity","doubly nonlinear system","singular potentials","bounded solutions","maximum principle","asymptotic limits","regularization"],"falsifier":"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).","tokens_in":22625,"feed_emoji":"🧪","tokens_out":13187,"duration_ms":126333,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bounded solutions proven for singular Cahn-Hilliard system","feed_subtitle":"No growth assumptions needed for existence; both regularization limits converge, one with explicit rate.","key_machinery":"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 $\\xi\\in\\beta(\\partial_t u)$.","core_discovery":"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): $\\|\\mu_\\delta-\\mu\\|_{L^2(0,T;V_0)}+\\|u_\\delta-u\\|_{H^1(0,T;H)}\\le M(\\delta^{1/4}+\\|u_{0\\delta}-u_0\\|_H+\\|g_\\delta-g\\|_{L^2(0,T;H)})$.","pith_inferences":["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 $[a'_0,b'_0]$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the δ=0 existence and uniqueness result and the estimates used to compare against when taking δ→0.","marker":"[4]"},{"why":"Provides the approximation scheme and preliminary estimates for the regularized system with ε,δ>0 that this paper extends to singular nonlinearities.","marker":"[5]"},{"why":"Gives the abstract doubly nonlinear discretization lemma invoked as Proposition 3.1, from which the maximum-principle boundedness follows.","marker":"[28]"},{"why":"Provides the convex-analysis tools for monotone graphs, resolvents, and smooth approximations used throughout the proofs.","marker":"[3]"},{"why":"Supplies the compactness criterion in L^p(0,T;B) used to extract convergent subsequences when passing to the ε→0 and δ→0 limits.","marker":"[31]"},{"why":"Source of the compactness-type estimate in Lemma 2.1 that controls L^2 norms against the dual-space norm and gradient norms.","marker":"[19]"}],"fun_headline_variants":["Bounded Cahn-Hilliard solutions without growth assumptions","New existence proof for doubly nonlinear Cahn-Hilliard","Singular potentials allowed in Cahn-Hilliard existence theorem","Cahn-Hilliard limits converge for both regularization parameters","Doubly nonlinear Cahn-Hilliard: boundedness and limit convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bounded Cahn-Hilliard solutions without growth assumptions","New existence proof for doubly nonlinear Cahn-Hilliard","Singular potentials allowed in Cahn-Hilliard existence theorem","Cahn-Hilliard limits converge for both regularization parameters","Doubly nonlinear Cahn-Hilliard: boundedness and limit convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1532,"prompt_tokens":1025,"completion_tokens":507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":641,"completion_tokens_details":{"reasoning_tokens":422}},"tokens_in":641,"tokens_out":507,"duration_ms":5485,"temperature":1.0,"reasoning_tokens":422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:55:12.332929+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"Bonetti, P","cited_arxiv_id":null,"evidence_quote":"Supplies the δ=0 existence and uniqueness result and the estimates used to compare against when taking δ→0."},{"cited_title":"Bonetti, P","cited_arxiv_id":null,"evidence_quote":"Provides the approximation scheme and preliminary estimates for the regularized system with ε,δ>0 that this paper extends to singular nonlinearities."},{"cited_title":"Nonlinear partial differential equation s with applications","cited_arxiv_id":null,"evidence_quote":"Gives the abstract doubly nonlinear discretization lemma invoked as Proposition 3.1, from which the maximum-principle boundedness follows."},{"cited_title":"Nonlinear differential equations of monotone types in Banach spaces","cited_arxiv_id":null,"evidence_quote":"Provides the convex-analysis tools for monotone graphs, resolvents, and smooth approximations used throughout the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the compactness criterion in L^p(0,T;B) used to extract convergent subsequences when passing to the ε→0 and δ→0 limits."},{"cited_title":"Quelques méthodes de résolution des probl èmes aux limites non linéaires","cited_arxiv_id":null,"evidence_quote":"Source of the compactness-type estimate in Lemma 2.1 that controls L^2 norms against the dual-space norm and gradient norms."}],"review_version":1}