{"id":"0a6d17d6-ecdb-4c5f-bcb6-9cd0cbccb8ad","arxiv_id":"2509.17710","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Two computable conditional a posteriori error estimators for a Keller-Segel FV-FE scheme are proven, and satisfaction of their conditions certifies existence of a weak solution up to time T.","lead":"This paper derives conditional a posteriori error estimates for a finite volume-finite element scheme for the Keller-Segel chemotaxis model, proving that when the scheme's residual conditions are met, a true weak solution exists on the simulated time interval. The work offers a template for turning simulations into rigorous existence certificates for blow-up-prone PDEs, though the conditions are currently practical only for very short times in two dimensions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'fully computable' condition in Theorem 6.7 rests on an unproved maximum-norm bound for a vector elliptic problem in Remark 3.4, not just on rough constants; without a valid explicit C∞, condition (7) cannot be verified and no executable a posteriori existence instance exists.","rationale":"The reader's weakest assumption correctly points to the non-explicit constants, especially C∞, and to the lack of documentation in the numerical experiments. My stress test concentrates on a more specific structural weakness underneath that same concern: the maximum-norm a posteriori bound for ||∇c̃||_L∞ is transferred to a vector-valued elliptic problem with nonstandard boundary data, and the transfer is neither proved nor fully specified. This strengthens rather than replaces the reader's conditional verdict. The main conditional existence architecture—combining the stability frameworks, residual estimates, and the blow-up criterion—remains plausible, but the advertised 'fully computable' condition is not presently established, because the key upper bound for the non-computable a(t) term is not justified in the manuscript. The proposed numerical test on a smooth manufactured solution would settle whether the vector maximum-norm bound is valid and, if it fails, would force the authors either to supply a correct bound or to weaken the claim of executable a posteriori existence. The verdict stays CONDITIONAL, hence 'UNCHANGED' relative to the reader.","tokens_in":29449,"tokens_out":15866,"duration_ms":147742,"concrete_test":"For Ω=(0,1)^2 (or the 3D slab used in §7.2) with ρ̄ = 1 + cos(πx)cos(πy), compute the exact solution c̄ of (I−Δ)c̄ = ρ̄ with homogeneous Neumann data, hence w = ∇c̄ exactly. Formulate a standard finite-element approximation q_h of w with the stated boundary condition ∇c̄·n = 0, and evaluate whether ||w − q_h||_L∞ ≤ C∞η∞ holds with the η∞ from Remark 3.4 and an explicit candidate C∞. If no well-posed discrete problem for q_h can be specified, or if the inequality fails, or if the scalar estimator in [19] cannot be applied componentwise without extra boundary/curl terms, then Remark 3.4 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires condition (7) (or (9)) to be checked from computable quantities only. In Theorem 6.7, the coefficient a(t) uses ||q_h(t)||_L∞ + C∞η∞(t) as an upper bound for ||∇c̃(t)||_L∞. Remark 3.4 asserts this by transferring a scalar maximum-norm a posteriori estimator from [19] to the vector-valued problem (I−Δ)∇c̄ = ∇ρ̄ with boundary condition ∇c̄·n = 0 on ∂Ω. The transfer is not proved: the discrete space for q_h, its boundary conditions, and the norm used for jumps of ∇q_h are never specified. For a scalar estimator to apply componentwise, each component of ∇c̄ would need scalar boundary data; on flat faces the normal component satisfies a Dirichlet condition, the tangential components Neumann conditions, and edge/corner behavior in polygonal domains is not discussed. Moreover, C∞ is only said to be estimable by inspecting [19, Lemma 6] and is never given, while the numerical experiments do not document how ||∇c̃||_L∞ was bounded. If Remark 3.4's bound is invalid or merely non-explicit, conditions (7)/(9) are not fully computable, so the claimed executable a posteriori existence conclusion lacks a verified premise. This is a correctness gap in the central theorem as stated, not only an issue of constants being too rough.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives two conditional a posteriori error estimates for a cell-centered finite volume / P1 finite element approximation of the parabolic-elliptic Keller-Segel system in two and three dimensions. The numerical density is reconstructed with a Morley-type interpolant, the chemical concentration with an elliptic reconstruction, and the residual is estimated in L2(0,T;H^-1). Two stability frameworks are combined with these residual bounds: one based on a Generalized Gronwall Lemma and one based on a local-in-time continuation argument. The central result, Theorem 6.7, states that if a fully computable condition involving the computable residual estimator and a maximum-norm bound on the chemical gradient is satisfied, then the a posteriori error estimator bounds the error to the exact solution, and consequently a weak solution exists up to time T. Numerical experiments in 2D compare the availability times of the two estimators, and experiments in 3D report convergence rates of the residual estimator and the reconstruction error under mesh refinement.","tokens_in":29931,"tokens_out":7262,"duration_ms":69391,"significance":"If the fully computable conditions are rigorously established, this is a valuable contribution to computer-assisted existence for a PDE with finite-time blow-up: the conditional inference from numerical behavior to existence is conceptually sound, and the paper correctly avoids circularity by computing the residual and the stability condition from the reconstruction alone. The paper also provides reproducible code and explicit comparisons of two stability frameworks. The residual estimates and the Morley-type reconstruction are worked out in considerable detail. However, the central claim of 'fully computable' conditions is not currently supported: the maximum-norm a posteriori bound for the vector-valued elliptic problem is asserted but not proved, the constant C∞ is never made explicit, and the unisolvence of the Morley-type element is left unproved. In addition, the numerical validations solve non-homogeneous manufactured problems whose source terms are not accounted for in the theoretical residual estimates. These are load-bearing gaps in the manuscript as it stands.","major_comments":[{"comment":"The claim that ||∇c̄(t)||L∞ ≤ ||qh(t)||L∞ + C∞η∞(t) is a computable upper bound is not established. The vector-valued problem (I−Δ)∇c̄ = ∇ρ̄ with ∇c̄·n = 0 is not the scalar reaction-diffusion problem covered by [19, Lemma 6], and the manuscript does not specify the finite element space for qh, the boundary conditions imposed on qh, or the definition of the jumps of ∇qh on a polygonal domain. Componentwise application of a scalar maximum-norm estimator is delicate because the normal and tangential components of ∇c̄ satisfy different boundary conditions on flat boundary parts. Moreover, C∞ is never given; the statement that it 'can be estimated explicitly by investigating the proof of [19, Lemma 6]' is not an explicit constant. Since a(t) in conditions (7) and (9) contains ||qh(t)||L∞ + C∞η∞(t), these conditions are not fully computable as written, and the a posteriori existence conclusion in Theorem 6.7 lacks a verified premise.","section":"Remark 3.4 / Theorem 6.7"},{"comment":"The unisolvence of the Morley-type element (K, P_K, Σ_K) is asserted with only 'Proof ... omitted for brevity', and no reference is given that covers this exact triple. The reconstruction ρ̃ is defined through (12)-(13) as the unique element of U_h, so a failure of unisolvence would invalidate the definition of ρ̃ and hence all subsequent residual estimates. A complete proof or a precise reference for this specific element is required.","section":"Section 5, Lemma 5.1"},{"comment":"The numerical experiments use manufactured solutions that solve non-homogeneous systems with nonzero source terms f and g, whereas Algorithm 4.3, the residual estimate (18), and Theorem 6.7 concern the homogeneous Keller-Segel system (KS.1)-(KS.3). The manuscript does not state how f and g are incorporated into the numerical scheme or into the residual estimator. Unless the residual and the stability framework are extended to include these source terms, the convergence results in Table 2 and the availability horizons in Table 1 do not provide evidence for Theorem 6.7 as stated.","section":"Sections 7.1-7.2, Eqs. (19)-(22)"},{"comment":"The existence inference in Corollary 3.6 is stated as a direct consequence of Proposition 2.5, but the proof is not given. Since Theorem 3.3 is a stability estimate for weak solutions that are already assumed to exist, the argument should explicitly apply the estimate on [0,T'] for all T'<Tmax, use the fact that the condition (7) and the constants remain controlled as T'→Tmax, and then pass to the limit to contradict the blow-up criterion. This is likely fixable, but it is not written out and is needed for the conditional existence claim.","section":"Corollary 3.6 / Proposition 2.5"}],"minor_comments":[{"comment":"The abstract contains a typo: 'a weak solution exits' should read 'a weak solution exists'.","section":"Abstract"},{"comment":"In the final display of part (ii), the error bound is written with ∥ρ(t,·)−¯ρ(t,·)∥, but the reconstruction is denoted ˜ρ throughout; this appears to be a typo and should be corrected to ∥ρ(t,·)−˜ρ(t,·)∥.","section":"Section 6.2, Theorem 6.7(ii)"},{"comment":"The constant C_Ω is described only by 'depends only on the shape parameter c_usr of S_T_h, see [20, Section 5.6.2.2]'. Since the theorem claims fully computable bounds, an explicit expression or a precise derivation for C_Ω should be given.","section":"Section 6.1, Lemma 6.3"},{"comment":"The claim about optimal orders of convergence of the Morley interpolation for d=2 is stated without proof or reference; if it is not needed for the main result, it could be removed or justified.","section":"Section 5, Remark 5.2"}],"recommendation":"major_revision","confidential_remarks":"The core idea is attractive and the conditional existence mechanism is sound, but the manuscript currently overstates what is fully computable. The missing proof of the maximum-norm estimator transfer and the missing explicit constant C∞ are load-bearing; the omitted unisolvence proof is also structural. The manufactured-solution experiments should be reconciled with the homogeneous theory or the theory extended. I would support reconsideration after these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main thing you should know: this is a serious attempt to turn a good simulation into a rigorous existence proof for the Keller-Segel system, and the conditional logic is basically right. The new work is the residual analysis for a cell-centered FV scheme coupled to a P1-FE scheme on well-centered tetrahedral meshes, using a Morley-type reconstruction and a dual/intersected mesh. That is a real step beyond the Cartesian and dG settings in earlier papers. The numerical tests in 3D are honest: the residual estimator scales at the expected order, and the authors openly admit that in 3D the Sobolev constants are too rough to make the conditions practical. That kind of candor is welcome.\n\nThe soft spots. The biggest one is Remark 3.4. The coefficient a(t) in Theorem 6.7 needs a computable upper bound for ||∇c~||_{L∞}, and the paper proposes to get it from a scalar maximum-norm a posteriori estimator in [19], applied to the vector-valued problem (I−Δ)∇c = ∇ρ with the boundary condition ∇c·n = 0. As written, the transfer is not justified: no discrete space for q_h, no boundary conditions, no componentwise argument, and the constant C∞ is never stated. This is not a minor annoyance about rough constants. Without a valid explicit C∞, condition (7) is not fully computable, so the central 'a posteriori verifiable existence' claim has no proved executable instance. The numerical experiments sidestep the issue: in the 2D availability test they use a manufactured solution and presumably the exact ||∇c||∞, so the problematic term is never actually bounded in practice.\n\nSecond, Lemma 5.1, the unisolvence of the Morley-type element, is dismissed with 'we omit it for brevity.' It is probably true and standard, but it is load-bearing. Third, some details of the estimators Θ_{K,n} are only described as 'summing the bounds' rather than written out, which will make replication harder but is not fatal.\n\nThe conditional existence inference itself is not circular. The conditions are evaluated from the reconstruction alone, and the blow-up criterion supplies the contradiction. So the architecture of the argument holds up; the gaps are in the computability layer. I would send this to peer review, but the referees should insist that Remark 3.4 be either proved properly or replaced by a different bound, and that Lemma 5.1 be proved or cited with a precise statement.\n\nThe paper is for people working on a posteriori analysis and computer-assisted proofs for nonlinear PDEs. With the computability gaps closed it would be a solid contribution.","headline":"A serious conditional a posteriori existence framework for Keller-Segel, with real novelty in the residual analysis, but the advertised fully computable condition depends on an unproved vector maximum-norm bound and an unstated constant.","tokens_in":30321,"tokens_out":5354,"would_cite":true,"duration_ms":49607,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M15","65M08","35K40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that conditional a posteriori error estimates for a finite volume scheme can certify existence of weak solutions to the Keller-Segel system up to the simulated time.","keywords":["conditional a posteriori error estimates","Keller-Segel system","finite volume scheme","a posteriori existence","weak solutions","blow-up","Morley-type reconstruction","finite element method"],"falsifier":"Run the two-dimensional manufactured example from Section 7.1 on successively finer meshes until condition (7) or (9) holds, and compare the certified bound ($\\delta A E$ or $\\Psi_{N_t-1}$) with the exactly known error $\\sup_t\\|\\rho-\\tilde\\rho\\|_{L^2}^2 + \\int_0^T |\\rho-\\tilde\\rho|_{H^1}^2\\,dt$; a single case in which the condition holds and the true error exceeds the bound would contradict Theorem 6.7.","tokens_in":29197,"feed_emoji":"🧮","tokens_out":12178,"duration_ms":92261,"temperature":0.7,"pith_summary":"This paper establishes conditional a posteriori error estimates for a finite volume-finite element discretization of the parabolic-elliptic Keller-Segel system, a model of chemotaxis. The central claim is that when a fully computable smallness condition holds for the reconstructed numerical solution, the estimator controls the error to the true weak solution in $L^\\infty(0,T;L^2)$ and $L^2(0,T;H^1)$, and as a consequence the weak solution actually exists on $[0,T]$. This converts a good simulation into a rigorous existence proof for a model whose solutions can blow up in finite time. The authors provide two stability frameworks, one based on an Adjusted Generalized Gronwall Lemma and one based on a local-in-time continuation argument, and they show numerically that the second condition is satisfiable on longer time horizons in two dimensions. The practical reach is limited by the sharpness of explicit constants, especially in three dimensions.","feed_headline":"A good simulation can prove Keller-Segel existence","feed_subtitle":"Two computable conditions let a finite-volume run certify a weak solution up to time T.","key_machinery":"The load-bearing object is the reconstructed pair $(\\tilde\\rho,\\tilde c)$: a Morley-type interpolant (a nonstandard $C^0$ finite element whose degrees of freedom are vertex values and face-normal flux integrals) of the cell-centered finite-volume density, linearly interpolated in time, together with $\\tilde c$ defined as the exact elliptic solution with right-hand side $\\tilde\\rho$, so that no residual enters the elliptic equation. Stability is supplied by two frameworks: an Adjusted Generalized Gronwall Lemma (Proposition 3.1) and a local-in-time continuation argument, each converting a small residual into an error bound under a smallness condition. The non-computable term $\\|\\nabla\\tilde c(t)\\|_{L^\\infty}$ is replaced by a computable upper bound using a $P_1$ finite element approximation $q_h$ plus a maximum-norm a posteriori estimator with explicit constant $C_\\infty$ (Remark 3.4). The residual is decomposed into diffusive, temporal, and convective parts, each estimated elementwise using explicit constants from Poincaré, trace, Sobolev-embedding, and elliptic-regularity inequalities.","core_discovery":"The paper's central claim is Theorem 6.7: for the Morley-type reconstruction $(\\tilde\\rho,\\tilde c)$ of the finite-volume/finite-element approximation, the full residual in $H^{-1}$ is bounded by a computable estimator that is a sum over mesh elements and time steps (18). If either the Gronwall-type condition (7) or the local-continuation condition (9) holds, then $\\sup_{t\\in[0,T]}\\|\\rho(t)-\\tilde\\rho(t)\\|_{L^2}^2 + \\int_0^T |\\rho(t)-\\tilde\\rho(t)|_{H^1}^2\\,dt$ is bounded by $\\delta A E$ or by $\\Psi_{N_t-1}$, respectively. By Proposition 2.5, that boundedness forces $T<T_{\\max}$, so in the paper's phrasing '$(\\rho,c)$ is a weak solution up to time $T$'. Existence is thereby inferred a posteriori from the numerical approximation rather than assumed a priori.","pith_inferences":["Beyond the paper, condition (9) could serve as a run-time certificate in three dimensions: if an explicit $C_\\infty$ and sharper Sobolev constants are supplied, verifying the condition on a feasible mesh would settle existence for concrete initial data whose long-time behavior is analytically open.","The same 'norm-blow-up criterion plus computable conditional estimator' template seems transferable to other nonlinear evolution equations with finite-time blow-up, such as semilinear heat equations, whenever a stability framework and a flux-preserving reconstruction are available.","A natural testable extension not implemented here is adaptive mesh refinement driven by the elementwise estimators $\\Theta_{K,n}$; refining where the estimator is large may extend the certified time horizon more efficiently than uniform refinement.","Using higher-order reconstructions or sharper stability constants would likely enlarge the attainable horizons more than raw mesh refinement, since the conditions in (7) and (9) depend exponentially on the stability coefficient through $E$."],"forward_implications":["When condition (7) holds for a computed run, the error to the exact weak solution is bounded by $\\delta A E$ and, by Corollary 3.6, the weak solution exists beyond the simulated time $T$.","When condition (9) has a root on every time step, the same conclusion holds with the local estimator $\\Psi_{N_t-1}$; in the paper's two-dimensional tests this condition is satisfied on substantially longer time horizons than the Gronwall condition.","Because the residual estimator and the factor $A$ shrink under mesh refinement for a convergent scheme, the conditional estimates become available on sufficiently fine meshes.","In the three-dimensional manufactured test, the residual estimator scales approximately linearly in the mesh size, matching the order of the $H^1$ error that the estimator bounds."],"supporting_citations":[{"why":"Supplies the existence and regularity theorem for weak solutions (Theorem 2.2) that the a posteriori existence argument starts from.","marker":"[9]"},{"why":"Supplies the blow-up criterion (Lemma 2.4) and an earlier conditional a posteriori framework for Keller-Segel that is extended here.","marker":"[28]"},{"why":"Supplies the Generalized Gronwall Lemma that is adjusted and used in Proposition 3.1.","marker":"[3]"},{"why":"Supplies the local-in-time continuation argument on which the second stability framework is based.","marker":"[14]"},{"why":"Supplies the Morley-type interpolation and flux-preservation properties used to reconstruct the numerical density and estimate residuals.","marker":"[47]"},{"why":"Supplies the maximum-norm a posteriori estimator used to make the infinity-norm term $\\|\\nabla\\tilde c\\|_{L^\\infty}$ computable via the constant $C_\\infty$.","marker":"[19]"},{"why":"Supplies the standard finite-element a posteriori estimates used for the elliptic reconstruction error $\\eta_\\Omega^n$.","marker":"[54]"},{"why":"Supplies an explicit Poincaré constant used in the sharpened two-dimensional constants.","marker":"[6]"}],"fun_headline_variants":["Simulation can certify Keller-Segel existence","A posteriori check proves Keller-Segel solution","Finite volume scheme yields existence proof","Keller-Segel existence via numerical test","Error estimator implies weak solution exists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole method depends on explicit, sufficiently sharp upper bounds for the Sobolev embedding constants, the elliptic regularity constant, and the maximum-norm constant $C_\\infty$; in three dimensions the known bounds are too rough for the conditions to hold on feasible meshes, and without an explicit $C_\\infty$ the term $\\|\\nabla\\tilde c(t)\\|_{L^\\infty}$ in the stability coefficient is not computable.","fun_headline_variants_meta":{"raw":{"variants":["Simulation can certify Keller-Segel existence","A posteriori check proves Keller-Segel solution","Finite volume scheme yields existence proof","Keller-Segel existence via numerical test","Error estimator implies weak solution exists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1374,"prompt_tokens":854,"completion_tokens":520,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":453}},"tokens_in":470,"tokens_out":520,"duration_ms":4833,"temperature":1.0,"reasoning_tokens":453,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:48:06.656307+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the two-dimensional manufactured example from Section 7.1 on successively finer meshes until condition (7) or (9) holds, and compare the certified bound ($\\delta A E$ or $\\Psi_{N_t-1}$) with the exactly known error $\\sup_t\\|\\rho-\\tilde\\rho\\|_{L^2}^2 + \\int_0^T |\\rho-\\tilde\\rho|_{H^1}^2\\,dt$; a single case in which the condition holds and the true error exceeds the bound would contradict Theorem 6.7.","supporting_citations":[{"cited_title":"Existence and nonexistence of solutions for a model of grav- itational interaction of particles, I","cited_arxiv_id":null,"evidence_quote":"Supplies the existence and regularity theorem for weak solutions (Theorem 2.2) that the a posteriori existence argument starts from."},{"cited_title":"A posteriori error control for a discontinuous Galerkin approximation of a Keller-Segel model","cited_arxiv_id":null,"evidence_quote":"Supplies the blow-up criterion (Lemma 2.4) and an earlier conditional a posteriori framework for Keller-Segel that is extended here."},{"cited_title":"Bartels.Numerical methods for nonlinear partial differential equations","cited_arxiv_id":null,"evidence_quote":"Supplies the Generalized Gronwall Lemma that is adjusted and used in Proposition 3.1."},{"cited_title":"Adaptivity and blow-up detection for nonlinear evolution problems","cited_arxiv_id":null,"evidence_quote":"Supplies the local-in-time continuation argument on which the second stability framework is based."},{"cited_title":"A Posteriori Error Estimations of Some Cell Centered Finite Volume Meth- ods for Diffusion-Convection-Reaction Problems","cited_arxiv_id":null,"evidence_quote":"Supplies the Morley-type interpolation and flux-preservation properties used to reconstruct the numerical density and estimate residuals."},{"cited_title":"Maximum-norm a posteriori error estimates for singu- larly perturbed elliptic reaction-diffusion problems","cited_arxiv_id":null,"evidence_quote":"Supplies the maximum-norm a posteriori estimator used to make the infinity-norm term $\\|\\nabla\\tilde c\\|_{L^\\infty}$ computable via the constant $C_\\infty$."}],"review_version":2}