{"id":"18567697-50ee-43aa-9f8d-f5c3667ebe2f","arxiv_id":"2608.05940","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors establish well-posedness of the outer chemotaxis-Euler system and of the boundary layer profiles arising in the ε→0 limit of the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity under Navier-slip conditions.","lead":"This paper derives and analyzes the boundary layer equations for a 2D chemotaxis-fluid system where bacteria sense oxygen through a logarithmic singularity, under Navier-slip boundary conditions. It proves local well-posedness for the zero-viscosity outer system and for the inner layer profiles, laying groundwork for a vanishing-viscosity convergence proof in Part II.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1 is asserted but not proven: Section 4 gives only a priori estimates, and the invoked contraction mapping requires a difference estimate that is absent and likely needs more v-regularity than the stated m≥2.","rationale":"The reader's verdict of CONDITIONAL is appropriate, but the most load-bearing weakness is not the formal ansatz (2.3), which the paper explicitly defers to Part II for convergence. The theorems of Part I are well-posedness statements for the outer and profile equations, and they stand or fall on the proofs of Theorems 2.1 and 2.2. The proof of Theorem 2.1 is the weakest link: Section 4 establishes a priori bounds for smooth solutions, but the existence step is delegated to an unspecified contraction mapping. The difference estimates needed for a contraction are nontrivial for this coupled hyperbolic-parabolic system with a characteristic boundary, and the term identified above suggests a possible derivative loss at the stated H^2 regularity for v. This is a concrete, checkable gap rather than a matter of taste. The reader's rationale did mention that the existence proof is sketched, so there is partial agreement, but the reader's formal weakest_assumption focused on ansatz validity, which I do not regard as load-bearing for the internal claims of this paper. Therefore I recommend keeping the CONDITIONAL verdict, with the condition being a complete contraction-mapping proof or a corrected regularity assumption in Theorem 2.1.","tokens_in":66190,"tokens_out":35342,"duration_ms":340186,"concrete_test":"Write out the missing contraction mapping for Theorem 2.1 at the minimal regularity m=2. Take two smooth solutions with data of equal size, form (δn, δv, δu), and derive the H^2×H^2×H^3 difference energy estimate. Isolate the term I = ∫ ∇^2δu · ∇^3 v_2 · ∇^2δv and the boundary term ∫ δ(n v_2) · ∇^2δn|_y=0. Determine whether I is bounded by the available norms ∥δu∥_{H^3}, ∥δv∥_{H^2}, ∥v_2∥_{H^2}, and ∥u_2∥_{H^3} without assuming H^3 regularity of v. If no cancellation is found, Theorem 2.1 needs v ∈ H^3 (or a different existence argument), and the m≥2 statement should be weakened; if a curl-free cancellation closes the estimate, record it explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 proves uniform a priori estimates for smooth solutions, but the final proof of Theorem 2.1 consists of one sentence: 'together with the contraction mapping argument and difference energy estimates, we establish Theorem 2.1.' No contraction map, no function-space setting, and no difference estimate is written. This matters because (2.4) is a coupled hyperbolic-parabolic system with characteristic boundary u_2=0, and local well-posedness does not follow formally from single-solution estimates. For two solutions, the difference δv satisfies ∂_t δv + ∇(u_1·δv + δu·v_2) − ∇δn = 0. After applying two derivatives and testing against ∇^2δv, the term ∫ ∇^2δu · ∇^3 v_2 · ∇^2δv arises (with the analogous boundary trace term in the n-equation). With only v_2 ∈ H^2, the case m=2 of Theorem 2.1, ∇^3 v_2 is not in L^2, so the difference estimate does not close at the stated regularity. The paper does not exhibit the cancellation that would make this term controllable. Since Theorem 2.2 consumes outer regularity from Theorem 2.1, this gap is load-bearing: if the contraction argument cannot be closed without v ∈ H^3, the theorem as stated is too strong. The ansatz-validity issue deferred to Part II is not a load-bearing premise for the well-posedness theorems themselves, which are stated for the profile and outer equations independently.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is described as Part I of a two-part study of boundary layers for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity in the half-space. After the Cole-Hopf transformation (1.2), the authors write down a matched asymptotic expansion (2.3) in powers of sqrt(ε), formally derive the leading-order outer system (2.4) and a hierarchy of inner profile equations (2.5)-(2.15), and state two main theorems: local well-posedness of the supercritical chemotaxis-Euler system (2.4) with data in H^m x H^m x H^{m+1}, m ≥ 2 (Theorem 2.1), and weighted anisotropic Sobolev regularity and uniqueness for the boundary-layer profiles (Theorem 2.2). Section 3 contains the formal expansion, Section 4 provides a priori estimates for smooth solutions of (2.4), and Section 5 proves weighted energy estimates for the linearized profile equations. The paper does not address convergence of the approximate solutions; that is explicitly deferred to Part II.","tokens_in":66487,"tokens_out":7458,"duration_ms":78858,"significance":"If the two theorems are fully proved, the paper would be a useful technical foundation for the boundary-layer analysis of a singular chemotaxis-fluid system: the profile equations are derived in detail, the weighted estimates for the z-dependent transport terms are nontrivial, and the regularity hierarchy in Theorem 2.2 is explicitly organized. The paper is also honest in stating that the actual convergence/error analysis is deferred to Part II. The main strength is the systematic weighted anisotropic energy framework for the inner-layer equations with unbounded coefficients. The main weakness is that the existence part of Theorem 2.1 is not written down, and the linear profile problems in Section 5 are treated by a priori estimates rather than by a constructed existence argument. There are no fitted parameters and the estimates are deterministic, which is a positive feature. In its current state, the manuscript is a set of substantial technical estimates rather than a completed well-posedness proof.","major_comments":[{"comment":"The proof of the central existence and uniqueness statement is a single sentence: \"together with the contraction mapping argument and difference energy estimates, we establish Theorem 2.1.\" No contraction map, no solution space, and no difference estimate is written. This is load-bearing because Lemmas 4.4, 4.5 and Corollary 4.1 are a priori estimates for a given smooth solution and do not by themselves produce a solution. Moreover, for the difference of two solutions, δv satisfies ∂_t δv + ∇(u_1·δv + δu·v_2) − ∇δn = 0; after two derivatives and testing against ∇^2δv, the term ∫ ∇^2δu · ∇^3 v_2 · ∇^2δv arises, together with an analogous boundary trace term in the n-equation. Since Theorem 2.1 assumes only v_2 ∈ H^2 in the case m = 2, ∇^3 v_2 is not in L^2, so the difference estimate does not close at the stated regularity unless a cancellation is exhibited. Unless the contraction argument is supplied with m ≥ 3 or an additional cancellation is shown, Theorem 2.1 is not established as stated.","section":"Section 4, Proof of Theorem 2.1"},{"comment":"These propositions are phrased as \"Then the solution f_B satisfies...\" and are the basis for concluding that each profile equation \"admits a unique solution\" in Lemmas 5.1-5.6. However, the proofs are weighted energy estimates for a putative solution; no existence argument is given. The model problems (5.1) and (5.20) are linear parabolic equations with the unbounded transport term z a(t,x)∂_z f and, in (5.20), nonhomogeneous boundary data ∂_z g_B|_{z=0} = µ. Existence and uniqueness do not follow automatically from the displayed estimates. The paper should either provide a semigroup, Galerkin, or approximation argument, or cite a theorem that covers this unbounded-coefficient setting. As written, the uniqueness assertions in Theorem 2.2 are not proved.","section":"Section 5, Propositions 5.1 and 5.2"},{"comment":"Theorem 2.1 states m ≥ 2, but Lemma 4.4 treats H^2, Lemma 4.5 treats H^3, and Corollary 4.1 starts its induction at m ≥ 4. The proof of Theorem 2.1 gives no explanation of how the intermediate cases 2 ≤ m ≤ 3 are covered, nor which regularity and compatibility conditions the contraction mapping would require. At minimum, the statement should be restricted to the range for which the a priori estimates are actually proved, or the missing argument should be supplied.","section":"Section 4, m-range in Theorem 2.1"}],"minor_comments":[{"comment":"There are several typographical issues, including \"chemotacxis\" in the abstract, \"Lebegue spaces\" in Section 2.1, and \"eatimate\" in Section 5; a careful copy-edit is needed.","section":"Abstract and Introduction"},{"comment":"The inductive statement (4.54) uses the shorthand H_m in some places and H_mxy in others; the norms should be written consistently throughout the proof.","section":"Corollary 4.1"},{"comment":"The formal derivation is very long and many displayed identities are verified only by phrases such as \"due to the rapid decay\"; a table or diagram showing which profile feeds into which equation would substantially improve readability.","section":"Section 3"},{"comment":"The restrictions m ≥ 11 and 7 ≤ m1 ≤ m-4 are stated but not derived in the proof of Theorem 2.2; the proof ends with a one-line reference to the preceding lemmas. A short derivation of these index restrictions should be added.","section":"Theorem 2.2 and Remark 2.2"},{"comment":"The paper repeatedly states that the convergence analysis and higher-order profiles are deferred to Part II; this should be explicitly identified as a limitation of Part I in the introduction, since the abstract's phrase 'boundary layer convergence' could be misread as a result established here.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the missing contraction proof for Theorem 2.1. If the authors can close the difference estimate at m = 2, or alternatively weaken the theorem to m ≥ 3, and can provide an existence basis for the linear profile problems in Propositions 5.1 and 5.2, the technical contribution would be substantial. The paper's fit with the journal depends on Part II delivering the promised convergence; Part I alone is a preparatory technical work. The self-citation [21] appears to be a comparable recent paper, and I see no circularity in the derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is worth reading if you work on chemotaxis-fluid boundary layers. It is the first to carry out a matched asymptotic expansion for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity under Navier-slip conditions, extending Hou-Wang's Keller-Segel boundary layer analysis to the fluid-coupled case. The formal expansion in Section 3 is intricate and self-consistent, and the weighted Sobolev estimates for the inner profiles in Section 5 are substantial. The local well-posedness of the supercritical chemotaxis Euler system is a natural target, and the a priori estimates in Lemmas 4.4-4.5 and Corollary 4.1 are detailed and appear correct.\n\nThe main soft spot is Theorem 2.1. The proof is one sentence: \"together with the contraction mapping argument and difference energy estimates, we establish Theorem 2.1.\" No contraction map or difference estimate is written. For a coupled hyperbolic-parabolic system with characteristic boundary u_2=0, single-solution estimates do not formally imply local well-posedness. The stress-test note identifies a plausible obstruction at the stated regularity m≥2: for two solutions, the difference equation produces a term involving ∇^3 v_2 against products of second derivatives, and ∇^3 v_2 is not in L^2 for v_2∈H^2. If that term cannot be cancelled or otherwise controlled, the theorem as stated is too strong. The paper does not show the cancellation. Since Theorem 2.2 consumes the outer regularity from Theorem 2.1, this gap is load-bearing.\n\nA second, lesser issue: the ansatz (2.3) is assumed; convergence of the approximate solution is deferred to Part II. That is explicitly acknowledged, so it is a limitation rather than a defect for the profile well-posedness claims.\n\nThe algebraic derivations in Sections 3 and 5 were spot-checked, not fully verified; they look plausible, and I did not find an obvious contradiction. The citation pattern is fine; citing [21] as comparable work is reasonable, and the central results are not derived from it.\n\nWho is this for? Researchers in boundary layer theory for chemotaxis or related hyperbolic-parabolic systems. It deserves a serious referee: the topic is important, the formal machinery is well-developed, and the gap in Theorem 2.1 is fixable if the difference estimates can be closed, possibly by requiring v∈H^3 or adding a cancellation argument. I would send it to review, with a request that the referee pay close attention to the contraction step.\n\nBest,\n[You]","headline":"Technically rich boundary-layer program for the chemotaxis-Navier-Stokes system; the profile estimates are solid, but the Euler well-posedness proof is missing its contraction argument and may require more regularity.","tokens_in":67028,"tokens_out":2543,"would_cite":false,"duration_ms":26540,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q92","76N05","35B44","35Q30","35Q35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity and Navier-slip boundary conditions, the leading outer system — the supercritical chemotaxis Euler equations — is locally well-posed, and every boundary-layer profile…","keywords":["Chemotaxis-Navier-Stokes system","Navier-slip boundary condition","Boundary layer","Logarithmic sensitivity","Well-posedness","Asymptotic expansion","Supercritical chemotaxis Euler equations","Weighted anisotropic Sobolev spaces"],"falsifier":"A direct numerical experiment on the $\\varepsilon>0$ system with smooth, compatible data: compute the weighted $L^2$ error between $n^\\varepsilon$ and the truncated expansion $n_{I,0}+\\sqrt{\\varepsilon}\\,n_{B,1}+\\varepsilon\\,n_{B,2}$ at a fixed time as $\\varepsilon\\to 0$; if the error does not vanish at the expected rate, the matched-expansion ansatz (2.3) is not the true structure of the solution, and Theorems 2.1-2.2 would describe profiles that do not arise in the actual limit.","tokens_in":65978,"feed_emoji":"🦠","tokens_out":10347,"duration_ms":102288,"temperature":0.7,"pith_summary":"This paper studies the two-dimensional chemotaxis-Navier-Stokes system with logarithmic sensitivity and small viscosity, under Navier-slip boundary conditions, and aims to establish the boundary-layer structure it develops as the viscosity tends to zero. Using the Cole-Hopf transformation $v=-\\nabla\\ln c$, the authors write solutions as matched asymptotic expansions in powers of $\\sqrt{\\varepsilon}$, coupling an outer part governed by the Euler-like limit with an inner part living in the stretched variable $z=y/\\sqrt{\\varepsilon}$. The paper proves two well-posedness results: the leading outer system (the supercritical chemotaxis Euler equations) has a unique local classical solution, and each derived boundary-layer profile equation has a unique solution in weighted anisotropic Sobolev spaces. These results give the analytical foundation for the boundary-layer convergence analysis announced for Part II, and they show that the near-wall distributions of fluid velocity, bacterial density, and oxygen concentration are determined jointly by viscous and chemotactic effects.","feed_headline":"Chemotaxis boundary layer and Euler limit are proven well-posed","feed_subtitle":"The viscous inner profiles and the outer ε=0 equations each have unique solutions, making the vanishing-viscosity limit tractable.","key_machinery":"The central object is the matched asymptotic expansion (2.3), which splits each unknown into an outer profile and an inner boundary-layer profile in $z=y/\\sqrt{\\varepsilon}$, together with the Cole-Hopf transformation $v=-\\nabla\\ln c$ that removes the logarithmic singularity. For the outer system, the load-bearing identity is the div-curl structure $\\nabla\\times v=0$, $\\nabla\\cdot u=0$, which yields the elliptic estimate $\\|\\nabla^{k+1}u\\|_{L^2}\\le C\\|\\nabla^k\\omega\\|_{L^2}$ and the cancellation $\\int \\partial_t v\\cdot(\\nabla\\partial_t v)\\cdot u=0$ that controls the supercritical nonlinear terms. For the inner profiles, the workhorse is the weighted anisotropic Sobolev space $H^{k,m,l}$ with norm $\\|(1+z^{2k})^{1/2}\\partial_x^\\alpha\\partial_z^\\gamma g\\|_{L^2}$, together with the observation that $z(1+z^{2\\kappa_{\\alpha,j}})\\le C(1+z^{2\\kappa_{\\alpha-\\beta,j}})$, which allows the unbounded factor $z$ to be absorbed by assigning larger polynomial weights to lower-order tangential derivatives.","core_discovery":"The central claim is that the matched asymptotic expansion is not merely formal: the leading outer profiles satisfy the chemotaxis-Euler system (2.4), which is locally well-posed for initial data in $H^m_{xy}\\times H^m_{xy}\\times H^{m+1}_{xy}$ satisfying compatibility and curl-free conditions; and the inner profiles $v_{B,0}^2, n_{B,1}, v_{B,1}^1, v_{B,1}^2, n_{B,2}, u_{B,1}^1, u_{B,2}^2, v_{B,2}^1, u_{B,2}^1, u_{B,3}^2, p_{B,2}$ defined by (2.5)-(2.15) each admit a unique solution with weighted anisotropic Sobolev regularity, provided the outer solution has sufficiently high tangential regularity. The main difficulties overcome are the loss of diffusion in the Euler limit, handled through the curl-free structure of $v$ and elliptic div-curl estimates, and the unbounded normal transport terms $z\\,a(t,x)\\partial_z f$ in the inner equations, handled through polynomial weights and boundary homogenization.","pith_inferences":["The same weighted-estimate cascade likely extends to all higher-order profiles $j\\ge 2$: since each new profile solves a linear equation whose source terms are already controlled, an induction analogous to Corollary 4.1 should yield regularity with a loss of roughly three tangential derivatives per level, a claim the paper leaves implicit.","The curl-free hypothesis on $v_0$ is the structural hinge of Theorem 2.1: the cancellation that kills the worst supercritical terms uses $\\nabla\\times v_0=0$, so dropping it would likely destroy local well-posedness or force a much shorter time of existence; this suggests a natural test problem for ill-posedness.","The explicit integral formulas (2.6), (2.13)-(2.15) turn the boundary layer into a computable object: comparing direct simulations of the $\\varepsilon>0$ system against $n_{I,0}+\\sqrt{\\varepsilon}\\,n_{B,1}+\\varepsilon\\,n_{B,2}$ would give a quantitative check of the ansatz before Part II appears.","Because the leading inner corrections satisfy $n_{B,0}=0$ and $u_{B,0}=0$ while $v_{B,0}^2$ differs from the outer trace, the boundary layer at leading order is purely a vertical redistribution of the transformed chemotactic velocity; the cell density itself changes only at order $\\varepsilon^{1/2}$, which predicts a stratification of the oxygen gradient rather than of cell count near the wall."],"forward_implications":["Part II's convergence proof now has the ingredients it needs: the outer solution exists on $[0,T]$ with the stated Sobolev regularity, and every inner profile it feeds is a well-posed solution of its equation in the weighted spaces.","For data with $m\\ge 11$, the first several boundary-layer profiles through order $\\varepsilon$ in density and velocity and their integral corrections are uniquely determined, so the formal expansion identifies the actual asymptotic limit rather than a spurious candidate.","The regularity transfer $7\\le m_1\\le m-4$ quantifies how much outer regularity survives into the boundary layer: constructing the first-order inner profiles costs between three and four derivatives of the outer solution.","The supercritical chemotaxis Euler equations inherit a local well-posedness theory from the energy estimates: no diffusion is needed in the $v$-equation when the initial data are curl-free and compatible.","The Navier-slip boundary conditions translate into explicit boundary data for the inner profiles, such as $v_{B,0}^2|_{z=0}=-v_{I,0}^2$, so the boundary layer is forced by the mismatch between the outer trace and the slip condition, not by an arbitrary ansatz."],"supporting_citations":[{"why":"Proposes the bacterial chemotaxis-fluid model that (1.1) specializes, supplying the biological setting and the singular logarithmic sensitivity that the paper must handle.","marker":"[32]"},{"why":"Introduces the Cole-Hopf-type transformation $v=-\\nabla\\ln c$ that removes the logarithmic singularity and produces the transformed system (1.4) studied throughout.","marker":"[19,22]"},{"why":"Provides the matched-asymptotics derivation of boundary-layer profile equations for the Keller-Segel system in the half-plane, which the present paper extends to the Navier-Stokes-coupled case.","marker":"[12]"},{"why":"Gives the div-curl elliptic estimate in the half-plane used as Lemma 4.1 to control $u$-regularity through the vorticity $\\omega$.","marker":"[8]"},{"why":"Supplies the half-plane regularity framework from which the div-curl norm equivalence used in Lemma 4.1 is taken.","marker":"[33]"},{"why":"The Sobolev-space reference for the interpolation and trace/embedding inequalities (Lemma 4.2 and related) used throughout the outer-layer energy estimates.","marker":"[1]"},{"why":"The nearest existing boundary-layer result for chemotaxis-Navier-Stokes with Navier-slip conditions; the present work extends this line to logarithmic sensitivity and to the supercritical Euler limit.","marker":"[21]"}],"fun_headline_variants":["Boundary layer and Euler limit well-posed for chemotaxis","Well-posed inner profiles and chemotaxis-Euler limit","Exact expansion proves chemotaxis-Euler well-posedness","Logarithmic chemotaxis: boundary layer well-posed","Vanishing viscosity limit rigorous for chemotaxis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that the matched asymptotic expansion (2.3) genuinely describes solutions of the $\\varepsilon>0$ system—that is, that the true solution is an outer flow plus a rapidly decaying layer of width $\\sqrt{\\varepsilon}$; the proof of that convergence is deferred to Part II, so the profile equations studied here would be meaningless if the ansatz were not realized by the actual system.","fun_headline_variants_meta":{"raw":{"variants":["Boundary layer and Euler limit well-posed for chemotaxis","Well-posed inner profiles and chemotaxis-Euler limit","Exact expansion proves chemotaxis-Euler well-posedness","Logarithmic chemotaxis: boundary layer well-posed","Vanishing viscosity limit rigorous for chemotaxis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1381,"prompt_tokens":933,"completion_tokens":448,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":363}},"tokens_in":549,"tokens_out":448,"duration_ms":4880,"temperature":1.0,"reasoning_tokens":363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:38:51.431918+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical experiment on the $\\varepsilon>0$ system with smooth, compatible data: compute the weighted $L^2$ error between $n^\\varepsilon$ and the truncated expansion $n_{I,0}+\\sqrt{\\varepsilon}\\,n_{B,1}+\\varepsilon\\,n_{B,2}$ at a fixed time as $\\varepsilon\\to 0$; if the error does not vanish at the expected rate, the matched-expansion ansatz (2.3) is not the true structure of the solution, and Theorems 2.1-2.2 would describe profiles that do not arise in the actual limit.","supporting_citations":[{"cited_title":"Tuval, L","cited_arxiv_id":null,"evidence_quote":"Proposes the bacterial chemotaxis-fluid model that (1.1) specializes, supplying the biological setting and the singular logarithmic sensitivity that the paper must handle."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the matched-asymptotics derivation of boundary-layer profile equations for the Keller-Segel system in the half-plane, which the present paper extends to the Navier-Stokes-coupled case."},{"cited_title":"Gilbarg, N.S","cited_arxiv_id":null,"evidence_quote":"Gives the div-curl elliptic estimate in the half-plane used as Lemma 4.1 to control $u$-regularity through the vorticity $\\omega$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the half-plane regularity framework from which the div-curl norm equivalence used in Lemma 4.1 is taken."},{"cited_title":"Adams, J.J.F","cited_arxiv_id":null,"evidence_quote":"The Sobolev-space reference for the interpolation and trace/embedding inequalities (Lemma 4.2 and related) used throughout the outer-layer energy estimates."},{"cited_title":"Zero-viscosity limit of the chemotaxis-Navier-Stokes equations with the Navier-slip boundary condition","cited_arxiv_id":"2605.02394","evidence_quote":"The nearest existing boundary-layer result for chemotaxis-Navier-Stokes with Navier-slip conditions; the present work extends this line to logarithmic sensitivity and to the supercritical Euler limit."}],"review_version":1}