{"id":"af672862-f2cc-4005-90e8-ba9b1719b27f","arxiv_id":"2505.04503","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Local and global strong well-posedness, exponential stability, global boundedness, and positivity for Keller-Segel-Navier-Stokes and chemotaxis-consumption systems on bounded Lipschitz domains in critical Besov spaces.","lead":"Rough-walled containers can now be handled in the mathematics of bacteria swimming in fluid: this preprint proves unique strong solutions for the Keller-Segel-Navier-Stokes system on bounded Lipschitz domains, in critical Besov spaces, including global small-data solutions and exponential stability. The significance is that chemotaxis-fluid models are shown to be well-posed in realistic nonsmooth geometries, a barrier that previously required smooth boundaries.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3 omits the necessary smallness condition |u_s|≤δ; for f=0 it claims exponential stability of arbitrarily large constant equilibria that are linearly unstable.","rationale":"The reader's weakest-assumption concern was the external square-root characterizations on Lipschitz domains; that concern is reasonable but does not identify the concrete gap in the paper's own statements. The paper's Lemma 4.2 and Remark 4.3 show that |u_s| smallness is needed in the maximal-regularity perturbation argument, but Theorem 2.3's Assumption 2.2 does not state it. The f=0 case makes the omission visible: Assumption 2.2 is vacuous, large u_s is allowed, and the linearization at (u_s,u_s,0) has an unstable eigenvalue whenever u_s>λ_1+1. Thus the global-stability claim in Theorem 2.3 is false as stated. The local well-posedness theorem and the overall method appear sound, and the fix is likely simple — add |u_s|smallness to Assumption 2.2 and propagate it through Theorems 2.3, 2.4, and 7.1 — so conditional acceptance rather than outright rejection is the appropriate revised verdict.","tokens_in":27237,"tokens_out":23460,"duration_ms":227867,"concrete_test":"Fix any bounded Lipschitz domain Ω, let λ_1>0 be its first nonzero Neumann eigenvalue, choose u_s>λ_1+1, set f≡0, and take initial data u_0=u_s+εφ_1, v_0=u_s+εφ_1, w_0=0, where φ_1 is the first nonzero Neumann eigenfunction (smooth, mean zero, hence in the stated Besov spaces) and ε is small enough that κ in Theorem 2.3 is satisfied. Solve the linearized system at (u_s,u_s,0): the φ_1-mode growth rate is (2λ_1+1+√(1+4u_sλ_1))/2 > 0. If this positive growth rate is observed numerically or analytically, the exponential decay claimed in Theorem 2.3 cannot hold, confirming that Assumption 2.2 must include a condition such as |u_s|≤δ.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Assumption 2.2 imposes smallness only on the product |u_s|‖f‖ (with ‖f‖_{W^{-1,2}} for n=2 and ‖f‖_{L^{3/2}} for n=3), not on |u_s| itself. Lemma 4.2, however, requires |u_s|‖f‖_{L^2}+|u_s|≤δ, and Remark 4.3 keeps the +|u_s|≤δ term. For f≡0, Assumption 2.2 is vacuous, so Theorem 2.3 as stated allows arbitrarily large constant equilibria (u_s,v_s,w_s)=(u_s,u_s,0). Such equilibria need not be exponentially stable. Linearizing (KSNS) about (u_s,u_s,0) on a Neumann eigenfunction φ_k of −Δ with eigenvalue λ_k gives û_t = −λ_k û + u_s λ_k v̂ and v̂_t = −(λ_k+1)v̂ + û, i.e. a 2×2 matrix with trace −(2λ_k+1) and determinant λ_k(λ_k+1−u_s). If u_s > λ_1+1, the determinant is negative and one eigenvalue is positive, so the stationary solution is linearly unstable. Since f=0 satisfies Assumption 2.2 for any u_s, Theorem 2.3 asserts exponential decay for these data, contradicting the linearized instability. The proof gap is that Theorem 2.3 invokes Lemma 4.2, whose hypotheses include |u_s|≤δ, without transferring this condition into Assumption 2.2.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a maximal-regularity framework for the Keller-Segel-Navier-Stokes system (KSNS) and the chemotaxis-consumption variant (CCNS) on bounded Lipschitz domains in dimensions 2 and 3. The main results are: local strong well-posedness for initial data in critical Besov spaces (Theorem 2.1); global strong well-posedness and exponential stability of nontrivial stationary solutions under a smallness condition (Theorem 2.3); global boundedness and positivity for smoother data (Theorem 2.4); and analogous statements for CCNS (Theorem 7.1). The proofs rely on square-root domain characterizations for the Neumann Laplacian and Stokes operator on Lipschitz domains, time-weighted maximal regularity, and a parametric contraction argument.","tokens_in":27517,"tokens_out":9482,"duration_ms":90857,"significance":"If the stability statement is corrected, this would be a substantial contribution: it extends critical-space well-posedness for chemotaxis-fluid systems from smooth or convex domains to general bounded Lipschitz domains, and it uses the deep Lipschitz-domain operator theory in a careful and mostly explicit parameter regime. The smallness constants are universal rather than fitted, and the bilinear estimates are presented in detail. The local existence, boundedness, and positivity parts appear coherent and well supported by the cited operator-theoretic results. However, the global stability theorem as stated is false, so the paper cannot be accepted without amendment.","major_comments":[{"comment":"The global stability statement is false as stated because Assumption 2.2 does not control |u_s|. For f = 0 the assumption is vacuous, so the theorem applies to constant equilibria (u_s,u_s,0) for arbitrarily large u_s. Linearizing (KSNS) about such an equilibrium on a Neumann eigenfunction of -Δ with eigenvalue λ_k gives the system a' = -λ_k a + u_s λ_k b, b' = a - (λ_k+1)b, whose 2x2 matrix has determinant λ_k(λ_k+1-u_s). For any u_s larger than λ_k+1 for some Neumann eigenvalue λ_k, the determinant is negative and a positive eigenvalue exists, so the equilibrium is linearly unstable and cannot be exponentially stable. The proof gap is that Lemma 4.2, which the proof of Theorem 2.3 invokes, requires |u_s| ||f||_{L^2} + |u_s| ≤ δ, and Remark 4.3 explicitly retains the +|u_s| term, but this condition is never transferred to Assumption 2.2. The theorem must be amended by adding |u_s| ≤ δ (or the full sum) to Assumption 2.2 and by aligning the norms used there with those used in Lemma 4.2.","section":"Section 4, Theorem 2.3 (with Assumption 2.2 and Lemma 4.2)"},{"comment":"The missing smallness on |u_s| propagates to Theorem 2.4 and to Theorem 7.1(2)-(3), since both statements depend on the stationary construction and on the δ from Theorem 2.3. The same correction must be made consistently in these statements, otherwise the boundedness, positivity, and CCNS stability results inherit the counterexample described above.","section":"Theorems 2.4 and 7.1"}],"minor_comments":[{"comment":"The manuscript contains several typographical and OCR-like artifacts (e.g., 'NA VIER-STOKES' in the title, 'underyling', 'arbritray', 'ch emotaxis') that should be cleaned before publication.","section":"Throughout"},{"comment":"The opening phrase 'Let all parameters be chosen as in Theorems 2.1, 2.2, 2.3 and 2.4' refers to Assumption 2.2, which is not a theorem; please renumber to avoid confusion.","section":"Theorem 7.1"},{"comment":"In the interpolation equality following 'Using [39, Thm. 3.5]', the pair is written as (H^{s0,q}_0, H^{s0,q}_0) but should be (H^{s0,q}_0, H^{s1,q}_0).","section":"Section 8, proof of Proposition 8.1"},{"comment":"The assertion that the R-bound of f(z) = z^{1-β}(1+A)^β (νz + ν^{1-1/β}(1+A))^{-1} is independent of ν is stated without proof; a short homogeneity argument (e.g., setting w = z/(ν^{-1/β}(1+A))) would make the verification transparent.","section":"Lemma 3.4"},{"comment":"Remark 4.3 already contains the corrected condition with the +|u_s| term; the authors should either promote this condition to Assumption 2.2 or explain explicitly why the theorem's assumptions imply it.","section":"Remark 4.3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read.\n\nThe genuinely new thing is Theorem 2.1: local strong well-posedness for the KSNS and CCNS systems in critical Besov spaces on bounded Lipschitz domains. That is a real step beyond smooth-domain results, and the way the authors use square-root domain characterizations to get the interpolation spaces is elegant. The bilinear estimates in Lemmas 3.3 and 3.4 look right, modulo details a referee will want spelled out, and the local theory hangs together. I also think the treatment of the CCNS variant as a corollary is honest.\n\nThe soft spot is Theorem 2.3. The statement only assumes the product smallness in Assumption 2.2, i.e. |u_s| times a norm of f. But Lemma 4.2, which the proof invokes, requires |u_s| itself to be small: the condition there is |u_s|∥f∥_{L^2} + |u_s| ≤ δ. For f ≡ 0, Assumption 2.2 is vacuous. Take any large constant equilibrium (u_s, u_s, 0). Linearizing about this state on a Neumann eigenfunction of eigenvalue λ_k gives a 2×2 matrix with determinant λ_k(λ_k+1 − u_s), which is negative when u_s > λ_1+1. So the equilibrium is linearly unstable, yet Theorem 2.3 as written claims exponential stability for all f satisfying Assumption 2.2. That is not a nitpick; it is a load-bearing gap in the main stability statement. The fix looks straightforward—add |u_s| ≤ δ (or the version from Remark 4.3) to the assumptions of Theorem 2.3 and Theorem 7.1. The proof of Theorem 2.3 already says 'let δ be small enough that Lemma 4.2 applies', so the authors clearly intended that condition; they just omitted it from the statement. With that repair, the global and stability results should go through.\n\nSecond-tier concerns: Lemma 3.4 has a terse R-boundedness step (the independence of ν in the Kalton-Weis application) that needs more detail. The dependence on deep external results—square-root domains, maximal regularity on Lipschitz domains—is acceptable because they are published, but a referee must check the parameter ranges (2.7) carefully. The positivity argument in Section 5 is fine.\n\nBottom line: the paper deserves a serious referee. The local theory and the framework are valuable, and the stability gap is fixable. Whoever referees it should flag the missing |u_s| smallness and ask for a corrected statement. I would not desk-reject.","headline":"The local well-posedness framework on Lipschitz domains is genuinely new and mostly solid, but the stability theorem as stated is false for f=0 because it omits a necessary smallness condition on |u_s|.","tokens_in":28091,"tokens_out":4470,"would_cite":true,"duration_ms":43043,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35Q92","92C17"],"pacs":[],"model":"deepseek-v4-flash","headline":"On bounded Lipschitz domains in two and three dimensions, the Keller-Segel-Navier-Stokes system and its chemotaxis-consumption variant admit unique local strong solutions in critical Besov spaces, global strong solutions near equilibria…","keywords":["Keller-Segel-Navier-Stokes system","chemotaxis-consumption model","Lipschitz domains","critical Besov spaces","maximal L^p regularity","strong well-posedness","exponential stability","positivity"],"falsifier":"For a bounded Lipschitz domain with a reentrant corner, compute explicitly the domains D((−Δ_q)^{1/2}) and D($A^{{1/2}}$) for q satisfying the paper's admissible interval; if either is not the corresponding $W^{{1,q}}$ space, then the half-interpolation identification fails and the Besov-space trace computation in Theorem 2.1 does not follow.","tokens_in":27002,"feed_emoji":"🦠","tokens_out":9650,"duration_ms":92905,"temperature":0.7,"pith_summary":"This paper proves that the Keller-Segel-Navier-Stokes system, which couples bacterial chemotaxis to an incompressible buoyancy-driven fluid, is strongly well-posed on bounded Lipschitz domains in dimensions two and three. For initial data in scaling-critical Besov spaces, a unique local strong solution exists. Near nontrivial equilibria, sufficiently small data produce global solutions that converge exponentially to the equilibrium, and for smoother data the solutions are globally bounded in space and time while preserving nonnegativity of the population density. The same results are established for the chemotaxis-consumption variant. The significance is that the theory works without any smoothness of the boundary, which matters for realistic containers and for domains with corners.","feed_headline":"Keller-Segel-Navier-Stokes models are well-posed on Lipschitz domains","feed_subtitle":"Local and global strong solutions, exponential stability, boundedness, and positivity hold in two and three dimensions.","key_machinery":"The argument recasts the system as a semilinear evolution equation ∂_t U + A U = Φ(U,U) on a product space whose components are a negative-order Sobolev space for the density, an L^q space for the chemoattractant, and the solenoidal L^q space for the fluid, with A a triangular matrix built from the Neumann Laplacian, a shifted Neumann Laplacian, and the Stokes operator. The load-bearing identity is the square-root domain characterization D((−Δ_q)^{1/2}) = $W^{{1,q}}$(Ω) and D($A^{{1/2}}$) = $W^{{1,q}}$_{0,σ}(Ω), valid for q in a narrow interval controlled by a geometric constant that depends on the Lipschitz boundary. This identifies the half-interpolation space as L^q × $W^{{1,q*}}$ × $W^{{1,q}}$_{0,σ}, which makes the bilinear estimates for the nonlinearity Φ possible through Sobolev embeddings. Time-weighted maximal L^p regularity in the critical-space framework turns those estimates into a contraction argument; for the global result, the linearization about a stationary solution is shown to inherit maximal regularity under a smallness condition on the product of the mean density and the coupling force.","core_discovery":"The central claim is that the Lipschitz geometry of the domain does not obstruct the strong well-posedness theory of these chemotaxis-fluid models. Theorem 2.1 states that for any bounded Lipschitz domain in two or three dimensions, any admissible exponents p and q, and any coupling force f in L^n, every initial datum in the critical Besov spaces $B^{{n/q−2}}$_{q,p,0}(Ω) × $B^{{n/q*}}$_{q*,p}(Ω) × $B^{{n/q−1}}$_{q,p,0,σ}(Ω) gives rise to a unique local strong solution in a time-weighted maximal-regularity space. Theorem 2.3 adds that if the data lie in a small ball around a stationary solution whose fluid part solves a stationary Navier-Stokes system driven by the mean population density times f, then the solution exists globally and converges exponentially to the equilibrium. Theorem 2.4 shows that sufficiently regular small data give solutions that are globally bounded in space and time and that a nonnegative initial density stays nonnegative. Theorem 7.1 transfers all of these statements to the chemotaxis-consumption-Navier-Stokes system.","pith_inferences":["The method suggests that other chemotaxis-fluid couplings with the same quadratic structure, for example models with logistic growth or altered boundary conditions, should inherit the same well-posedness pattern as long as the square-root domain characterizations remain available; this is an extension the paper does not pursue.","The admissible range of q is tied to a geometric constant of the Lipschitz domain, so the range of spaces in which the theory works may shrink as corners become sharper; checking whether the interval becomes empty for some domain would reveal whether the critical-space statement is optimal rather than purely technical.","Because positivity is proved through a linear comparison principle that is independent of the smallness condition, the positivity conclusion may extend to the local strong solutions of Theorem 2.1 whenever they exist, even far from equilibrium; this is a direct but unstated corollary of the proof structure."],"forward_implications":["Local strong well-posedness holds in scaling-critical Besov spaces, extending the critical-space theory of the Navier-Stokes equations to the coupled chemotaxis-fluid system on arbitrary bounded Lipschitz domains.","Nontrivial equilibria are exponentially stable in the critical functional setting whenever the data are close enough and the coupling force is small in the stated sense.","For smoother data, solutions are globally bounded in space and time and preserve positivity of the population density, making the solutions suitable for biological interpretation.","The same existence, stability, boundedness, and positivity results cover both the Keller-Segel-Navier-Stokes and the chemotaxis-consumption-Navier-Stokes systems, with arbitrary coupling forces such as buoyancy."],"supporting_citations":[{"why":"This is the external theorem identifying the domain of the square root of the Neumann Laplacian with W^{1,q}(Ω) on Lipschitz domains, the identity that anchors the whole interpolation computation.","marker":"[18]"},{"why":"This identifies the domain of the square root of the Stokes operator with W^{1,q}_{0,σ}(Ω) in two-dimensional bounded Lipschitz domains, the second anchor of the interpolation computation.","marker":"[11]"},{"why":"This provides the same square-root characterization for the Stokes operator in three-dimensional bounded Lipschitz domains, closing the three-dimensional case.","marker":"[36]"},{"why":"This supplies the L^p resolvent estimates for the Stokes operator on Lipschitz domains that fix the admissible range of q used throughout.","marker":"[34]"},{"why":"This supplies the boundary regularity theory for the Laplacian on Lipschitz domains used for embeddings of the Neumann-Laplacian domain into fractional Sobolev spaces.","marker":"[9]"},{"why":"This gives the H∞-calculus for the Stokes operator on bounded Lipschitz domains, from which the maximal L^p regularity of the linear operator is derived.","marker":"[21]"},{"why":"This provides the time-weighted maximal-regularity framework for quasilinear parabolic equations in critical spaces that the contraction argument uses.","marker":"[33]"},{"why":"This supplies the weighted Sobolev embeddings used in the bilinear estimate that is uniform in the time horizon.","marker":"[26]"},{"why":"This gives the interpolation formulas for Besov and Bessel-potential spaces used to identify the critical initial-data spaces in the theorems.","marker":"[39]"}],"fun_headline_variants":["Strong solutions for Keller-Segel-Navier-Stokes on Lipschitz domains","Chemotaxis-fluid systems well-posed on bounded Lipschitz domains","Lipschitz domains: strong solutions for chemotaxis-Navier-Stokes","Strong well-posedness for Keller-Segel-Navier-Stokes on Lipschitz domains"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the external result that, on bounded Lipschitz domains, the square roots of the Neumann Laplacian and the Stokes operator have first-order Sobolev spaces as their domains for q in a narrow interval set by the Lipschitz geometry; if that characterization fails for an admissible q, the interpolation computation and the entire contraction argument collapse.","fun_headline_variants_meta":{"raw":{"variants":["Strong solutions for Keller-Segel-Navier-Stokes on Lipschitz domains","Chemotaxis-fluid systems well-posed on bounded Lipschitz domains","Lipschitz domains: strong solutions for chemotaxis-Navier-Stokes","Strong well-posedness for Keller-Segel-Navier-Stokes on Lipschitz domains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00116,"raw_usage":{"total_tokens":4782,"prompt_tokens":902,"completion_tokens":3880,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":3792}},"tokens_in":518,"tokens_out":3880,"duration_ms":26752,"temperature":1.0,"reasoning_tokens":3792,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:27:18.530489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a bounded Lipschitz domain with a reentrant corner, compute explicitly the domains D((−Δ_q)^{1/2}) and D($A^{{1/2}}$) for q satisfying the paper's admissible interval; if either is not the corresponding $W^{{1,q}}$ space, then the half-interpolation identification fails and the Besov-space trace computation in Theorem 2.1 does not follow.","supporting_citations":[{"cited_title":"Jerison and C","cited_arxiv_id":null,"evidence_quote":"This is the external theorem identifying the domain of the square root of the Neumann Laplacian with W^{1,q}(Ω) on Lipschitz domains, the identity that anchors the whole interpolation computation."},{"cited_title":"Gabel and P","cited_arxiv_id":null,"evidence_quote":"This identifies the domain of the square root of the Stokes operator with W^{1,q}_{0,σ}(Ω) in two-dimensional bounded Lipschitz domains, the second anchor of the interpolation computation."},{"cited_title":"Tolksdorf","cited_arxiv_id":null,"evidence_quote":"This provides the same square-root characterization for the Stokes operator in three-dimensional bounded Lipschitz domains, closing the three-dimensional case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This supplies the L^p resolvent estimates for the Stokes operator on Lipschitz domains that fix the admissible range of q used throughout."},{"cited_title":"Fabes, O","cited_arxiv_id":null,"evidence_quote":"This supplies the boundary regularity theory for the Laplacian on Lipschitz domains used for embeddings of the Neumann-Laplacian domain into fractional Sobolev spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This gives the H∞-calculus for the Stokes operator on bounded Lipschitz domains, from which the maximal L^p regularity of the linear operator is derived."},{"cited_title":"Pr¨ uss, G","cited_arxiv_id":null,"evidence_quote":"This provides the time-weighted maximal-regularity framework for quasilinear parabolic equations in critical spaces that the contraction argument uses."},{"cited_title":"Meyries and M","cited_arxiv_id":null,"evidence_quote":"This supplies the weighted Sobolev embeddings used in the bilinear estimate that is uniform in the time horizon."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This gives the interpolation formulas for Besov and Bessel-potential spaces used to identify the critical initial-data spaces in the theorems."}],"review_version":1}