{"id":"dce163ca-43df-4574-9ac3-15463ab3da0e","arxiv_id":"2506.18617","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove well-posedness and new regularity results for a convective bulk-surface Cahn-Hilliard system under Leray-type velocity fields, using a new elliptic regularity theory for singular nonlinearities.","lead":"This paper improves the mathematical theory of a model that describes how two fluids mix or separate, both in the bulk and on the container boundary. It proves that the model still has unique solutions when the fluid velocity is only as regular as in the standard Navier-Stokes theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's CONDITIONAL verdict centers on the domination assumption (2.4) and deferrals to prior work. I examined the most sensitive point of the proof—the elliptic regularity theory in Sections 4–5 and its use in Theorem 3.6. The central theorem is explicitly conditional on (2.4), which is a genuine scope restriction but not a correctness flaw. The standard logarithmic potential satisfies (2.4), so the primary physical model remains covered. I checked the sign-sensitive boundary estimate (5.8): for α<0, the monotonicity of s↦F1'(αh_k(s))|F1'(αh_k(s))|^{p−2} combined with the factor αv−u gives a nonnegative product, so the displayed inequality is valid. No circularity appears in the uniform estimates: Step 4 closes via (7.36) and Step 6 via Gronwall, with norms of the velocity fields and initial data on the RHS. The convergence statement in Step 7 claiming weak-* L∞(0,T;H3) is stronger than the available L2(0,T;H3) bound; however, the theorem only needs weak L2(0,T;H3) convergence, which follows from (7.72), so this is a harmless typo. The abstract's omission of compatibility condition (C) and the slightly overbroad uniqueness corollary in Theorem 3.5 are presentation issues rather than mathematical errors, since Theorem 3.6's velocity class is strong enough to satisfy the stated uniqueness hypotheses. Overall, the central claim holds under the stated assumptions, and the CONDITIONAL verdict—pending fuller details of deferred results—remains appropriate.","tokens_in":47052,"tokens_out":43010,"duration_ms":356583,"concrete_test":"Re-derive the boundary estimate in Proposition 5.3 for both signs of α∈[-1,1], tracking the constants in (5.8) with the Young alternative |αv−u|≤C; then re-run the Yosida-regularized version (Remark 5.6) for the logarithmic potential to confirm the constants in (5.9) remain independent of λ and α. If they do, the uniform estimates in Step 6 of Theorem 3.6 hold; if not, the strong regularity proof has a hidden dependence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The strong well-posedness claim (Theorem 3.6) is explicitly conditional on assumptions (2.4), (C), constant mobilities, K∈[0,∞), L∈(0,∞], and Ω∈C3; within these assumptions the proof is coherent. The domination assumption (2.4) is used only under that hypothesis, and the standard logarithmic potential satisfies it with κ1=1. The apparent sign issue for α<0 in (5.8) actually closes because the monotone composition f∘(αh_k) has the same sign behavior as α(v−α−1u), making the boundary term non-positive. Minor overstatements—abstract omitting (C) and Theorem 3.5's uniqueness corollary requiring a slight adaptation for purely L2 velocities—do not affect the main result. The only notable typo is the unsupported 'weakly-* in L∞(0,T;H3)' convergence in Step 7, which is stronger than the established L2(0,T;H3) bound but is not needed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a convective bulk-surface Cahn–Hilliard system with dynamic boundary conditions and singular potentials, for prescribed divergence-free velocity fields. The main results are: (i) existence of weak solutions when the velocities satisfy only L^2(0,T;L^2_div)×L^2(0,T;L^2_τ) regularity, improving earlier L^3/L^{2+ω} assumptions; (ii) a continuous-dependence and uniqueness estimate measured in L^2 for velocity differences; and (iii) existence of unique strong solutions, with higher regularity, for Leray-type velocities in L^∞(0,T;L^2)∩L^2(0,T;H^1). The proofs are based on a new regularity theory for a bulk-surface elliptic system with singular nonlinearities, which is developed in Section 5 via subdifferential techniques and a series of L^p estimates. The paper is carefully organized, but two of the three central proofs are only outlined and refer to the authors' earlier work, and one displayed identity in the proof of Theorem 3.6 is not justified.","tokens_in":47239,"tokens_out":29069,"duration_ms":260290,"significance":"If the results are correct, they constitute a meaningful improvement over the state of the art: the Leray-type velocity regularity is the natural class arising from weak solutions of Navier–Stokes equations, and the weak well-posedness under purely L^2 velocities is relevant for coupled bulk-surface flow models. The auxiliary elliptic theory of Section 5, including the maximal monotonicity of the subdifferential, the L^p estimates, and the higher Sobolev regularity for the singular bulk-surface elliptic system, is a substantive contribution that may be of independent interest. The paper provides explicit a priori estimates, including estimates with constants independent of the final time, and clearly states all hypotheses such as the compatibility condition (C) and the domination assumption (2.4). The main concern is that Eq. (7.54), which is load-bearing for the proof of Theorem 3.6, appears to be false as stated, and the proofs of Theorem 3.2 and Theorem 3.5 rely heavily on references to prior work without a complete derivation of the modified estimates.","major_comments":[{"comment":"Equation (7.54) is not justified and appears to be false in general. In the simplified case v=w=0, F=G=0, L=K=∞, the displayed equality would assert ||∂tφ||²_{L2} = −C⟨µ,∂tφ⟩. For a Neumann eigenfunction with −∆φ=λφ, one has ∂tφ=−λ²φ and µ=λφ, so ||∂tφ||²_{L2}=λ⁴||φ||²_{L2} but −⟨µ,∂tφ⟩=λ³||φ||²_{L2}, which are not equal for λ≠1. Since this equality is used to derive the differential inequality (7.60) and hence the estimates (3.17)–(3.18), the proof of Theorem 3.6 contains a load-bearing gap; a corrected bound for the F''_2-terms (e.g., via (7.50)–(7.52) with a suitable absorption argument) should be supplied.","section":"Section 7, Eq. (7.54)"},{"comment":"The proofs of the two main weak well-posedness results are largely delegated to the authors' earlier papers. The proof of Theorem 3.2 states that the uniform estimates follow by repeating [34, Theorem 3.4]; since the new point is precisely the weaker velocity class, the convective estimates should be shown in detail. The proof of Theorem 3.5 reduces the key differential inequality (6.2) to 'repeating the line of argument' in [35] and [34], and the displayed estimate (6.3) contains an unsquared ||(v,w)||_{L2} term. Please provide complete arguments (or a detailed appendix) for these two load-bearing results.","section":"Section 6, proofs of Theorem 3.2 and Theorem 3.5"}],"minor_comments":[{"comment":"The velocity space in the statement contains a duplicated L2(0,T): 'let (v,w)∈ L2(0,T ; L2(0,T ; L2_div(Ω)× L2_τ(Γ)))' should be 'let (v,w)∈ L2(0,T; L2_div(Ω)× L2_τ(Γ))'.","section":"Theorem 3.2"},{"comment":"The convergence '(µλ,θλ)→(µ,θ) weakly-* in L∞(0,T;H3)' is not supported by the established L2(0,T;H3) bound; it should read 'weakly in L2(0,T;H3)' unless an additional L∞(0,T;H3) bound is proved.","section":"Section 7, Step 7"},{"comment":"In the first term of the final bound in (6.3), ||(v,w)||_{L2} should be squared to match the use of Young's inequality.","section":"Section 6, Eq. (6.3)"},{"comment":"The boundary term in (4.12) appears to have an extra factor α: after combining the boundary contributions from the bulk and surface equations it should be ∫Γ ∂n uλ (αG'_1,λ(vλ)−F'_1,λ(uλ)) dΓ. Please verify.","section":"Section 4, Eq. (4.12)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely correct in its broad strategy after the gap in Section 7 is fixed, and the auxiliary elliptic results are solid. My main concern is the degree of reliance on [34,35] for two of the three central theorems, together with the false identity in Eq. (7.54). These issues should be resolved in a revision; they do not appear to be insurmountable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a credible, incremental but genuinely useful extension of the authors' own earlier work [34,35]. The new content is: weak well-posedness when the prescribed velocity fields are only L^2 in space-time (Theorem 3.2), uniqueness/stability with the velocity difference measured in L^2 (Theorem 3.5), and strong well-posedness for Leray-type velocities (Theorem 3.6). The engine is a new well-posedness and elliptic regularity theory for the stationary bulk-surface system with singular nonlinearities (Section 5). That theory is the most valuable part and should survive as an independent tool.\n\nThe paper does its main job honestly. I checked the stress-test note carefully and it holds up: within the stated assumptions—constant mobilities, K in [0,infinity), L in (0,infinity], domain C^3, compatibility condition (C), and the domination assumption (2.4)—the proof is coherent. No essential gap jumps out. The domination assumption is a real restriction, but the standard logarithmic potential satisfies it with kappa1=1, so it is not an unreasonable price.\n\nSoft spots are real but moderate. The abstract overstates the scope: it omits the compatibility condition (C) and the restrictions on K and L needed for the strong result. Theorem 3.2's proof is not self-contained; it outlines a compactness argument from [34] and then uses the a posteriori separation to weaken the velocity assumptions. That is legitimate but leaves some work to the reader. Similarly, Theorem 3.5 says 'repeating the line of argument' from [35,34]—acceptable in a subfield where those arguments are well known, but a referee should ask for at least a sketch of the key Gronwall estimate. There is also a small technical slip in Step 7 of the proof of Theorem 3.6: the claim of weak-* convergence in L^infinity(0,T;H^3) is stronger than the established L^2(0,T;H^3) bound and is not needed; it should be corrected.\n\nAll told, the paper deserves a serious referee. It is not paradigm-shifting, but it removes a restrictive velocity regularity condition and supplies a reusable elliptic regularity theory. I would cite it if I were working on Navier-Stokes-Cahn-Hilliard systems.\n\nRecommendation: send it to review, with a request to fix the abstract's statement and to clarify the deferred arguments.","headline":"Solid, honest extension of the authors' own prior work; the new elliptic regularity theory is the real contribution.","tokens_in":47732,"tokens_out":2467,"would_cite":true,"duration_ms":23649,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K35","35D30","35A01","35A02","35Q92","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a bulk-surface Cahn–Hilliard system with singular potentials stays strongly well-posed when the driving velocity fields have only Leray-type regularity — the level weak solutions of the Navier–Stokes equations provide.","keywords":["convective Cahn-Hilliard equation","bulk-surface interaction","dynamic boundary conditions","Leray velocity fields","singular potentials","regularity theory","well-posedness"],"falsifier":"A concrete calculation settles the role of the weakest assumption: fix $\\alpha=1$, let $G_1$ be the logarithmic Flory–Huggins potential, for which $|G'_1(s)|$ grows like $|\\log(1-s)|$, and let $F_1$ satisfy (2.2)–(2.3) with $|F'_1(s)|\\sim(1-s)^{-\\gamma}$ for some $\\gamma\\in(0,1)$, so that (2.4) fails because $|F'_1(s)|/|G'_1(s)|\\to\\infty$ as $s\\to1$; then solve the elliptic system (5.1) with a family of smooth data $(f,g)$ and check whether the bound $\\|(F'_1(u),G'_1(v))\\|_{L^p}\\le C(1+\\|(f,g)\\|_{L^p})$ of Proposition 5.3 still holds. A sequence of data driving the left-hand side to infinity would show the regularity theory genuinely needs (2.4), while a verified bound would show the assumption can be weakened; in the admissible case (logarithmic potentials, Leray-type velocity fields, three dimensions), a high-resolution numerical solution of System (1.1) could confirm whether the claimed regularity $(\\mu,\\theta)\\in L^2(0,T;H^3)$ is actually realized.","tokens_in":46875,"feed_emoji":"🌊","tokens_out":34468,"duration_ms":258955,"temperature":0.7,"pith_summary":"This paper proves that a bulk-surface Cahn–Hilliard system with dynamic boundary conditions and singular double-well potentials — a diffuse-interface model of phase separation whose boundary carries its own Cahn–Hilliard-type equation — remains well-posed when it is driven by velocity fields much rougher than those previously allowed. Weak solutions are shown to exist for prescribed velocities merely in $L^2(0,T;L^2)$, down from the earlier requirement of $L^3$ bulk and $L^{2+\\omega}$ surface integrability, and the weak solution is unique and depends continuously on the data, with velocity differences measured only in the $L^2$ norm. The main result, Theorem 3.6, upgrades the solution to full strong regularity — $(\\partial_t\\phi,\\partial_t\\psi)\\in L^\\infty(0,T;(H^1_{L,\\beta})')\\cap L^2(0,T;H^1)$, $(\\phi,\\psi)\\in L^\\infty(0,T;W^{2,6})\\cap C(Q)\\times C(\\Sigma)$, $(\\mu,\\theta)\\in L^\\infty(0,T;H^1_{L,\\beta})\\cap L^2(0,T;H^3)$ — when the velocity fields have Leray-type regularity, the level that weak solutions of the Navier–Stokes equations supply. Since that is precisely the regularity a coupled fluid problem would hand to the Cahn–Hilliard subsystem, the result makes the model usable in bulk-surface Navier–Stokes–Cahn–Hilliard analysis. The proofs rest on a new well-posedness and regularity theory for a coupled bulk-surface elliptic system with singular nonlinearities, developed in Section 5, which the authors note may be of independent interest.","feed_headline":"Need only Leray-type flows for strong Cahn–Hilliard solutions","feed_subtitle":"The velocity assumption now matches what Navier–Stokes solutions actually deliver, easing coupling to fluid models.","key_machinery":"The load-bearing mechanism is a new regularity theory for the coupled bulk-surface elliptic system (5.1), $-\\Delta u+F'_1(u)=f$ in $\\Omega$, $-\\Delta_\\Gamma v+G'_1(v)+\\alpha\\partial_n u=g$ on $\\Gamma$, $K\\partial_n u=\\alpha v-u$ on $\\Gamma$, with singular convex potentials. Its three pillars are: the characterization of the subdifferential of the singular part of the free energy, $\\partial\\tilde{E}_K(u,v)=(-\\Delta u+F'_1(u),-\\Delta_\\Gamma v+G'_1(v)+\\alpha\\partial_n u)$, a maximal monotone operator whose essential domain is the pairs $(u,v)\\in H^2$ with $(F'_1(u),G'_1(v))\\in L^2$ satisfying the boundary condition (Proposition 4.2); the $L^p$ estimate $\\|(u,v)\\|_{W^{2,p}}+\\|(F'_1(u),G'_1(v))\\|_{L^p}\\le C(1+\\|(f,g)\\|_{L^p}+\\gamma(K)\\|\\partial_n u\\|_{L^p(\\Gamma)})$ for all $p\\in[2,\\infty]$, together with the separation property $|u|\\le 1-\\delta$, $|v|\\le 1-\\delta$ (Proposition 5.3); and the domination assumption (2.4), $|F'_1(\\alpha s)|\\le\\kappa_1|G'_1(s)|+\\kappa_2$, which is exactly what absorbs the boundary terms carrying the normal derivative and the trace of the bulk potential derivative in the estimates (5.8) and (5.20). These elliptic estimates are established uniformly in the Moreau–Yosida regularization parameter (a standard smoothing of the singular convex potentials), so the approximation scheme of Section 7 — regularized potentials, initial data defined through the elliptic system, and time-mollified velocity fields — produces the differential inequality (7.78) for $\\|(\\mu_\\lambda,\\theta_\\lambda)\\|^2_{L,\\beta}$, and Gronwall's lemma combined with elliptic regularity for bulk-surface systems delivers the regularity of Theorem 3.6.","core_discovery":"On the authors' own terms, the central result (Theorem 3.6) is the following. Let $\\Omega\\subset\\mathbb{R}^d$, $d=2,3$, be of class $C^3$, let the mobilities be constant, let $F_1,G_1$ be singular convex potentials satisfying (2.2)–(2.4), let $K\\in[0,\\infty)$ and $L\\in(0,\\infty]$, and let the initial data satisfy (3.1) together with the compatibility condition (C); if the prescribed velocity fields belong to the Leray class $(v,w)\\in L^\\infty(0,T;L^2_{\\rm div})\\cap L^2(0,T;H^1)$, and if $v|_\\Gamma=w$ almost everywhere in the case $K=0$, then the unique weak solution of System (1.1) obtained in Theorem 3.2 enjoys the regularity $(\\partial_t\\phi,\\partial_t\\psi)\\in L^\\infty(0,T;(H^1_{L,\\beta})')\\cap L^2(0,T;H^1)$, $(\\phi,\\psi)\\in L^\\infty(0,T;W^{2,6})\\cap C(Q)\\times C(\\Sigma)$, $(\\mu,\\theta)\\in L^\\infty(0,T;H^1_{L,\\beta})\\cap L^2(0,T;H^3)$, and $(F'(\\phi),G'(\\psi))\\in L^2(0,T;L^\\infty)\\cap L^\\infty(0,T;L^6)$, so that every equation of System (1.1) holds almost everywhere, with the explicit estimates (3.17)–(3.18). The paper also establishes weak well-posedness under the weaker velocity assumption (1.7), a continuous-dependence estimate (3.13) involving only the $L^2(0,T;L^2)$ norm of the velocity difference, and the higher time regularity $(\\phi,\\psi)\\in L^4(0,T;H^2)$ for $K\\in(0,\\infty)$ (with $L^3(0,T;H^2)$ for $K=0$) for every weak solution; by Remark 3.7(a), the strong result additionally covers $L=0$ in two dimensions, and in three dimensions covers $K=L=0$ under a compatibility condition on the regular parts of the potentials.","pith_inferences":["The restriction in Remark 3.7(a) — in three dimensions the case $L=0$ is covered only for $K=0$ together with a compatibility condition on the regular potentials — points to the trace relation $\\mu=\\beta\\theta$ on the boundary as the genuine obstacle: the approximation scheme must preserve that trace at the level of the regularized initial data, and Leray-type velocities do not supply the extra ti","Since the domination condition (2.4) enters only through the boundary-term absorptions (5.8) and (5.20), a natural weakening would allow $|F'_1(\\alpha s)|$ to grow like a power of $|G'_1(s)|$ plus a constant, with the exponent $p$ in (5.4) then depending on that power; testing this would show whether the restriction is physically necessary or an artifact of the method.","The stability estimate (3.13), which needs only the $L^2(0,T;L^2)$ norm of the velocity difference, is exactly the continuity statement that lets coupled schemes pass to the limit when velocities are recovered by compactness; the paper does not exploit this direction, but the estimate seems built for it.","The explicit bounds (3.17)–(3.18) grow exponentially in the accumulated $H^1$-norm of the velocity field; for velocities that decay in time, a Gronwall refinement should reduce this to a polynomial factor, which would matter for long-time and attractor analyses."],"forward_implications":["In any coupled bulk-surface Navier–Stokes–Cahn–Hilliard model, the velocity field delivered to the phase-field subsystem by the fluid's energy balance has exactly the Leray regularity required by Theorem 3.6, so the subsystem can be treated as strongly well-posed without demanding extra time regularity from the fluid; the paper identifies this as the main benefit over the earlier theory.","Weak well-posedness holds with velocity fields only in $L^2(0,T;L^2)$, and the stability estimate (3.13) measures velocity differences only in the $L^2(0,T;L^2)$ norm; both relaxations match what a compactness argument in a coupled scheme would supply.","Every weak solution gains the extra time regularity $(\\phi,\\psi)\\in L^4(0,T;H^2)$ for $K\\in(0,\\infty)$ (or $L^3(0,T;H^2)$ for $K=0$), and for $L\\in(0,\\infty]$ the weak solution satisfies the energy equality rather than merely an inequality.","The Section 5 elliptic theory — well-posedness, $W^{2,p}$ estimates, and separation from the pure phases, with constants independent of the smoothing parameter — applies to any Cahn–Hilliard- or Allen–Cahn-type system with dynamic boundary conditions and singular potentials, as the authors point out."],"supporting_citations":[{"why":"Prior weak- and strong-well-posedness results for this same model, which Theorems 3.2, 3.5 and 3.6 improve, and the source of an elliptic regularity proposition reused in Sections 5 and 7.","marker":"[34]"},{"why":"Introduced System (1.1) and the analysis of the regular-potential case; supplies the approximate-system existence and the chain rule invoked in the proof of Theorem 3.6.","marker":"[35]"},{"why":"The general well-posedness and regularity theory for bulk-surface elliptic systems — including the regularity theorem, the bulk-surface Poincaré inequality and the norm equivalence — on which Section 5 is built.","marker":"[32]"},{"why":"Establishes the domination inequality for the Yosida-regularized potentials, transferring assumption (2.4) to the approximations used in Sections 4–7.","marker":"[12]"},{"why":"The maximal-monotone-operator machinery used to prove the explicit subdifferential representation (4.1) and to pass to the limit in the Yosida approximation.","marker":"[7]"},{"why":"Supplies the absolute-continuity and chain-rule result for the singular energy, which yields the energy equality in Theorem 3.2.","marker":"[48]"},{"why":"Provides the trace interpolation inequality (Lemma 2.2) used to control the normal-derivative boundary terms in Corollary 5.4 and Proposition 5.5.","marker":"[47]"},{"why":"Provides the time-mollification of the velocity fields used in Step 3 of the proof of Theorem 3.6, with the convergence and norm bounds (7.15)–(7.16).","marker":"[3]"}],"fun_headline_variants":["Leray velocities suffice for strong Cahn-Hilliard solutions","Weaker velocity condition yields strong solutions","Cahn-Hilliard well-posed under Leray flows","Strong solutions with physical velocity fields","Leray-type flows ensure strong Cahn-Hilliard regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the domination condition (2.4) — that near the pure-phase values $\\pm 1$ the singular part of the bulk potential, evaluated at $\\alpha s$, cannot blow up faster than the singular part of the boundary potential at $s$ — because without that inequality the boundary terms carrying the normal derivative can no longer be absorbed, and the proof of Theorem 3.6 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Leray velocities suffice for strong Cahn-Hilliard solutions","Weaker velocity condition yields strong solutions","Cahn-Hilliard well-posed under Leray flows","Strong solutions with physical velocity fields","Leray-type flows ensure strong Cahn-Hilliard regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1988,"prompt_tokens":1193,"completion_tokens":795,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":809,"completion_tokens_details":{"reasoning_tokens":718}},"tokens_in":809,"tokens_out":795,"duration_ms":7326,"temperature":1.0,"reasoning_tokens":718,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:45:04.555208+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete calculation settles the role of the weakest assumption: fix $\\alpha=1$, let $G_1$ be the logarithmic Flory–Huggins potential, for which $|G'_1(s)|$ grows like $|\\log(1-s)|$, and let $F_1$ satisfy (2.2)–(2.3) with $|F'_1(s)|\\sim(1-s)^{-\\gamma}$ for some $\\gamma\\in(0,1)$, so that (2.4) fails because $|F'_1(s)|/|G'_1(s)|\\to\\infty$ as $s\\to1$; then solve the elliptic system (5.1) with a family of smooth data $(f,g)$ and check whether the bound $\\|(F'_1(u),G'_1(v))\\|_{L^p}\\le C(1+\\|(f,g)\\|_{L^p})$ of Proposition 5.3 still holds. A sequence of data driving the left-hand side to infinity would show the regularity theory genuinely needs (2.4), while a verified bound would show the assumption can be weakened; in the admissible case (logarithmic potentials, Leray-type velocity fields, three dimensions), a high-resolution numerical solution of System (1.1) could confirm whether the claimed regularity $(\\mu,\\theta)\\in L^2(0,T;H^3)$ is actually realized.","supporting_citations":[{"cited_title":"Knopf and J","cited_arxiv_id":null,"evidence_quote":"Prior weak- and strong-well-posedness results for this same model, which Theorems 3.2, 3.5 and 3.6 improve, and the source of an elliptic regularity proposition reused in Sections 5 and 7."},{"cited_title":"Knopf and J","cited_arxiv_id":null,"evidence_quote":"Introduced System (1.1) and the analysis of the regular-potential case; supplies the approximate-system existence and the chain rule invoked in the proof of Theorem 3.6."},{"cited_title":"Knopf and C","cited_arxiv_id":null,"evidence_quote":"The general well-posedness and regularity theory for bulk-surface elliptic systems — including the regularity theorem, the bulk-surface Poincaré inequality and the norm equivalence — on which Section 5 is built."},{"cited_title":"Calatroni and P","cited_arxiv_id":null,"evidence_quote":"Establishes the domination inequality for the Yosida-regularized potentials, transferring assumption (2.4) to the approximations used in Sections 4–7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The maximal-monotone-operator machinery used to prove the explicit subdifferential representation (4.1) and to pass to the limit in the Yosida approximation."},{"cited_title":"Rocca and G","cited_arxiv_id":null,"evidence_quote":"Supplies the absolute-continuity and chain-rule result for the singular energy, which yields the energy equality in Theorem 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the trace interpolation inequality (Lemma 2.2) used to control the normal-derivative boundary terms in Corollary 5.4 and Proposition 5.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the time-mollification of the velocity fields used in Step 3 of the proof of Theorem 3.6, with the convergence and norm bounds (7.15)–(7.16)."}],"review_version":2}