{"id":"27503c57-a9d5-433f-b5a9-cba0611b7800","arxiv_id":"2504.18484","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a total mixing condition (the initial density ratio has bounded variation), global weak solutions exist for the two-species cross-diffusion system with independent drifts and no self-diffusion.","lead":"This paper proves global existence of weak solutions for a two-species cross-diffusion system on a one-dimensional torus, with no self-diffusion and with independent drift potentials. It is the first such existence result covering mixed, non-segregated initial densities with differing drifts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The existence proof hinges on an unproved adaptation of Laborde's theorem to the torus; if the JKO/energy hypotheses do not transfer to (3.1), the regularised solutions underpinning all BV estimates and the limit passage do not exist.","rationale":"I followed the main chain: regularisation by (3.1), the change to (sigma, f(r)), the BV estimate via the dissipation of (3.8), the L^2-BV bound for sigma, the time-derivative bound, the Aubin-Lions-Simon compactness, and the passage to the limit. The sign-function manipulations in Theorem 3.5 are justified by Lemmas A.1-A.2 and the boundary-free integration on the torus; the product convergence in the cross-diffusion term is supported by the strong convergence of the bounded quotients and weak L^{3/2} convergence of derivatives. The apparent issue with quotients on a vacuum set is resolved by the strict positivity of sigma for t>0 from the Fokker-Planck Harnack estimate. The bootstrap in Section 4 is plausible. The single most load-bearing point, however, is Theorem 3.2: without existence of the regularised weak solutions, none of the subsequent smoothness or a priori estimates have a starting point. The proof of Theorem 3.2 is a citation plus an asserted domain adaptation, and Section 4 does not establish existence. This is exactly the weakest assumption identified by the Reader, and my review finds no independent fix for it in the manuscript. Therefore the verdict should remain CONDITIONAL: the central argument is coherent once the regularised solutions exist, but that existence is not fully demonstrated.","tokens_in":27646,"tokens_out":35342,"duration_ms":359155,"concrete_test":"Obtain Laborde [Lab20, Theorem 2.2] and check its hypotheses and proof steps for the system (3.1) with the energy E_eta on the flat torus. Specifically: (i) list every place the proof uses the no-flux boundary condition or the geometry of an interval; (ii) verify that replacing the interval by T preserves the JKO/optimal-transport steps; (iii) confirm that the constants eta and (1-2eta) keep the pressure law in the class treated (m=1). If (i)-(iii) cannot be verified, supply a direct existence proof for (3.1) (e.g., via a Galerkin or Schauder fixed-point argument) before the BV estimates can be used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 rests the whole approximation scheme on Theorem 3.2, whose proof is one paragraph: after checking that the energy E_eta is finite, the authors say only that 'it is simply left to consider the adaptation' of Laborde's Theorem 2.2 from a bounded interval with no-flux boundary conditions to the flat torus (lines around (3.1)-(3.2)). No JKO scheme, boundary-condition comparison, or coefficient correspondence is given. Section 4 (Propositions 4.1-4.3) upgrades weak solutions to smooth positive ones, so it presupposes existence. If Laborde's theorem relies on boundary terms in the Euler-Lagrange equations, on convexity of the domain, or on a pressure law that the factors eta and (1-2eta) alter, the existence of the regularised solutions rho_i^eta is not established. All a priori estimates (Theorem 3.5, Theorem 3.6, Corollary 3.7, Proposition 3.9) and the compactness argument in Theorem 2.2 are conditioned on that existence. This is a gap in the foundation of the central claim, not an internal contradiction after the regularised solutions are granted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the two-species cross-diffusion system (1.1) on the one-dimensional flat torus, with logarithmic pressure acting on the sum of the densities and with independent drift potentials. The main result (Theorem 2.2) asserts the global existence of weak solutions under hypotheses H1-H4, with the additional regularity rho_i in L^2([0,T];BV(T)) cap L^p_loc((0,T];BV(T)). The strategy is to pass to the variables (sigma, f(r)) with f(r)=log(rho_1/rho_2), regularize the system by adding linear self-diffusion (3.1), derive eta-independent BV estimates for f(r) and L^{3/2} estimates for partial_x sigma, and then obtain strong compactness of the individual densities via the Aubin-Lions-Simon lemma. Section 4 develops the smoothness and strict positivity of the regularized solutions, and Section 5 gives a conditional extension to reaction-cross-diffusion systems.","tokens_in":27871,"tokens_out":32222,"duration_ms":324725,"significance":"If the proof is completed, the result would be the first existence theorem for this type of cross-diffusion system allowing mixed initial data and independent drifts without self-diffusion. The change of variables to the logarithmic ratio and the BV energy-dissipation estimate for int |partial_x u| are genuinely novel, and the eta-independence of the constants in Theorem 3.5 is carefully tracked. The energy-dissipation estimate, the sigma-estimate, and the compactness argument are written in considerable detail and cohere. The paper is also honest in marking the reaction-term extension as conditional. However, the proof currently relies on an unproved adaptation of Laborde's existence theorem to the torus and on a regularity lemma whose proof is invalid as stated; both are load-bearing but appear repairable.","major_comments":[{"comment":"The existence of weak solutions to the eta-regularised system (3.1) on the torus is the foundation of the entire proof, but Theorem 3.2 is proved by asserting that [Lab20, Theorem 2.2] applies 'up to the constants eta and (1-2eta)' after a 'direct adaptation' from a bounded interval with no-flux boundary conditions to the flat torus. No JKO scheme, boundary-term comparison, or coefficient correspondence is provided, and the proof of Theorem 3.2 is only one paragraph. Because all a priori estimates (Theorems 3.5 and 3.6, Corollary 3.7, Proposition 3.9) and the compactness passage in Theorem 2.2 are conditional on the existence of these regularised solutions, this is a load-bearing gap. The authors should either state and prove the adapted existence theorem for the torus or give a detailed reduction from the interval case.","section":"Section 3, Theorem 3.2"},{"comment":"The proof of Proposition 4.1 contains a serious technical error. In the Duhamel formula (4.4), the divergence term has the wrong sign; the correct expression is ∂x[(ξ̃φ - 2ũ∂xφ)*2 Φ], not ∂x[(2ũ∂xφ - ξ̃φ)*2 Φ]. More importantly, the estimate of the initial-data term uses Young's inequality as if * were a space-time convolution, although * is defined as spatial convolution only. With u0∈L^p, the homogeneous heat flow e^{tΔ}u0 need not lie in L^p([0,T];W^{1,p}(T)) for p>3, so the proposition as stated is false for general L^p initial data. Since Proposition 4.3 uses Proposition 4.1 to bootstrap smoothness of the regularised solutions, this proof must be corrected, for instance by assuming u0∈W^{1,p} in the base case or by invoking standard maximal regularity with the appropriate initial-data space.","section":"Section 4, Proposition 4.1"}],"minor_comments":[{"comment":"The proof of the L^2 H^1 estimate for sqrt(rho_1+rho_2) is essentially a citation to [Lab20, Proposition 3.4]; since the regularised system (3.1) differs from the one in [Lab20] by the constants eta and (1-2eta), the authors should either reproduce the entropy-dissipation computation or state precisely the adapted estimate.","section":"Section 3, Theorem 3.6"},{"comment":"The adaptation of the Harnack inequality from [Bog+15, Theorem 8.1.3] to the flat torus is asserted after 'tracking constants' rather than proved; the global inequality (4.9) deserves a few lines of justification showing why periodicity turns the local estimate into a global one.","section":"Section 4, Proposition 4.2"},{"comment":"The abstract states that the main results 'naturally extend to similar systems involving reaction terms', but Theorem 5.1 is conditional on the existence of a suitable smooth approximation that is not constructed; the wording should be adjusted to avoid overclaiming.","section":"Abstract and Section 5"},{"comment":"The text says the time-derivative bound holds 'for any s>0 such that H^s(T)⊂W^{2,∞}(T)', but this embedding requires s>5/2; the condition should be stated precisely.","section":"Section 3, Proposition 3.9"},{"comment":"There are numerous typos and grammatical slips ('spacial', 'righter most', 'infimise', 'the term ... defines a weak solution'); a careful proofread is needed.","section":"Throughout"},{"comment":"The existence of the approximating sequence of smooth positive initial densities satisfying the uniform bounds in (3.30) is asserted without proof; a short mollification-and-normalisation argument would make this step transparent.","section":"Proof of Theorem 2.2"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The manuscript is promising and the main strategy is credible. The two major issues I identified are repairable: the torus adaptation of Laborde's theorem should be written out in detail, and Proposition 4.1 needs a correct proof or a strengthened hypothesis. I do not see grounds for rejection, but the paper should not be accepted in its present form. The reaction-term section is clearly labeled conditional, which is appropriate. No concerns about novelty or citation practices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper proves the first global existence result for a two-species cross-diffusion system on the 1D torus with independent drifts and no self-diffusion, under a total mixing condition. The central mechanism is new: writing the system in (σ, f(r)) with f = log(ρ1/ρ2), they show the BV norm of ∂x f(r)+V dissipates via a Grönwall argument, which bypasses the usual need for self-diffusion. The estimates are written out in detail and the limit passage is coherent. I read through the main body and found no fatal flaw in the a priori estimates or compactness argument.\n\nThe soft spots are real but not disqualifying. Theorem 3.2, the existence of solutions to the regularized system, is imported from Laborde and the adaptation from a bounded interval with no-flux boundary to the flat torus is asserted rather than proved. This is load-bearing: all subsequent estimates are for solutions of that regularized system. The adaptation is very likely routine—the JKO scheme on a compact manifold is easier than on a domain with boundary—but the authors should either prove it or state it as an explicit assumption. A referee should push for this. The abstract also claims the results 'naturally extend' to reaction terms, but the actual theorem (5.1) is conditional on a suitable smooth approximation that the paper does not construct, and Remark 1.3 admits this. That is an overstatement. There is also a missing figure reference.\n\nThe math, as far as I can verify, checks out. The citation pattern is honest; the paper credits Laborde, Kim–Mészáros, and the optimal transport literature appropriately. I did not independently verify every imported estimate from [Lab20] and [LSU68], but the use looks standard.\n\nThis paper is for PDE analysts working on degenerate cross-diffusion and JKO-based methods. It deserves a serious referee. My recommendation: accept for peer review, request a revision that fills the Laborde adaptation gap and softens the reaction-term claim.","headline":"A new existence proof for a degenerate cross-diffusion system with independent drifts, via a genuinely novel change of variables, but the paper leans on an unproved import from Laborde and overstates its reaction-term extension.","tokens_in":28429,"tokens_out":5537,"would_cite":false,"duration_ms":53845,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35B65","35K40","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves global existence of weak solutions for a two-species cross-diffusion system on the one-dimensional torus with independent drift potentials and no self-diffusion, provided the initial log-ratio has bounded variation.","keywords":["cross-diffusion system","drift-diffusion equation","total mixing","no self-diffusion","weak solutions","BV regularity","one-dimensional torus","logarithmic pressure law"],"falsifier":"Verify the proof of Theorem 3.2 directly on the torus: show that the cited existence theorem for the η-regularised system remains valid on R/Z with the same energy and constants. If there is a pair of initial data satisfying H1-H4 and potentials in $W^{{2,1}}$(T) with V1−V2 in $W^{{3,1}}$(T) for which the regularised system has no weak solution on the torus, the approximation step collapses and the central claim is unsupported. Equivalently, a counterexample to Theorem 2.2 satisfying H1-H4 would refute the claim.","tokens_in":1617,"feed_emoji":"🧮","tokens_out":7115,"duration_ms":131164,"temperature":0.7,"pith_summary":"The paper proves that a two-species cross-diffusion system on the one-dimensional torus, where diffusion acts only on the sum of the species and the two species feel independent drift potentials, has global weak solutions for any time horizon. The key step is a change of variables to the sum density and the logarithmic species ratio, which turns the difficult product term into a BV estimate on the ratio. Under a total-mixing condition on the initial data, which still allows vacuum and blow-up, the ratio remains of bounded variation for all times, so the two densities never form inter-species interfaces. The obtained solutions have each species in $L^{2}$([0,T];BV(T)) and in L^p_loc((0,T];BV(T)) for every finite p. The same strategy extends to reaction-cross-diffusion systems, conditional on a suitable smooth approximation.","feed_headline":"Global solutions found for cross-diffusion with independent drifts","feed_subtitle":"First existence result for mixed initial densities and differing drifts with no self-diffusion.","key_machinery":"The load-bearing device is the nonlinear change of variables $(\\rho_1,\\rho_2) \\mapsto (\\sigma,f(r))$ with $\\sigma=\\rho_1+\\rho_2$ and $f(r)=\\log(r/(1-r))=\\log(\\rho_1/\\rho_2)$. The function $f$ is chosen so that $f'(r)=1/(r(1-r))\\ge 4$, which allows $\\mathrm{BV}$ control of $f(r)$ to become $\\mathrm{BV}$ control of $r=\\rho_1/(\\rho_1+\\rho_2)$. The second engine is the energy $\\int_{\\mathbb{T}} |\\partial_x f(r)+V|\\,dx$, whose time dissipation is computed for the $\\eta$-regularised system and closed by a Grönwall argument; Kato's inequality and the absence of boundary on the torus make the dangerous second-derivative term vanish. The third ingredient is the family of regularised systems with an added linear self-diffusion $\\eta$, whose smooth, strictly positive solutions justify the computations and whose estimates are uniform in $\\eta$.","core_discovery":"The paper's central claim is Theorem 2.2: under hypotheses H1-H4 (the sum of the initial densities lies in $L\\log L$, the log-ratio $\\log(\\rho_{1,0}/\\rho_{2,0})$ lies in $\\mathrm{BV}(\\mathbb{T})$, the potentials satisfy $V_1,V_2\\in W^{2,1}(\\mathbb{T})$, and $V_1-V_2\\in W^{3,1}(\\mathbb{T})$) there exists a weak solution of the system $\\partial_t\\rho_i = \\partial_x(\\rho_i \\partial_x(\\log(\\rho_1+\\rho_2)+V_i))$ for $i=1,2$ on the flat torus, with $\\rho_1,\\rho_2 \\in L^2([0,T];\\mathrm{BV}(\\mathbb{T})) \\cap L^p_{\\mathrm{loc}}((0,T];\\mathrm{BV}(\\mathbb{T}))$ for every $p\\in [1,\\infty)$. The proof rewrites the system in the variables $\\sigma=\\rho_1+\\rho_2$ and $f(r)=\\log(\\rho_1/\\rho_2)$, where $r=\\rho_1/\\sigma$, and shows that $f(r)$ obeys a transport-type equation whose $\\mathrm{BV}$ norm is controlled through the dissipation of the energy $\\int_{\\mathbb{T}} |\\partial_x f(r)+V|\\,dx$, with $V=\\partial_x(V_1-V_2)$. Because $f'(r)=1/(r(1-r))\\ge 4$, the $\\mathrm{BV}$ bound on $f(r)$ transfers to $r$; combined with Sobolev estimates on $\\sigma$, this yields strong compactness of the individual densities and passage to the limit from the $\\eta$-regularised system. The paper states this to be the first existence result for mixed initial densities and differing drifts in the absence of self-diffusion.","pith_inferences":["If the cited existence theorem for the regularised system does not extend to the torus as asserted, the approximation chain could be repaired by a direct well-posedness proof for the regularised system on R/Z, but the paper does not supply that proof.","The BV-on-ratio mechanism may transfer to other pressure laws: any choice of f with f' bounded below would convert BV control of f(r) into BV control of r, so the logarithmic pressure law is likely not essential.","The dissipation calculation has the same flavour as standard Fokker-Planck energy estimates, so the method may yield quantitative rates or a variational rewriting of the system; this is speculative.","A numerical experiment on the torus could test whether the Grönwall bound on ∫|∂x f(r)+V| is sharp when V1−V2 has large oscillations in W^{3,1}."],"forward_implications":["Weak solutions exist for arbitrarily long time horizons, with each species spatially BV; if the initial sum lies in L^q for q>2, the solution gains L^q([0,T];BV(T)) regularity.","Total mixing is propagated in time: since log(ρ1/ρ2) stays BV, the two species cannot develop the sharp interfaces seen in segregated cross-diffusion models.","No ordering or joint structural condition on the drifts is required; only the stated Sobolev regularity of V1 and V2 is needed.","The same machinery works for reaction-cross-diffusion systems with bounded Lipschitz reaction terms, provided a suitable smooth approximation with uniformly positive densities exists.","The hypotheses allow vacuum and blow-up in the initial data, provided the two densities become asymptotically proportional near such points."],"supporting_citations":[{"why":"Supplies the existence theorem for the η-regularised system on which the approximation chain is based.","marker":"[Lab20]"},{"why":"Provides the Aubin-Lions-Simon compactness lemma used to pass to the limit in the individual densities.","marker":"[Sim87]"},{"why":"Provides the parabolic Lp estimates and bootstrapping argument for regularity of the regularised solutions.","marker":"[LSU68]"},{"why":"Supplies the Harnack inequality and the Lp_loc regularity result for the limiting Fokker-Planck equation.","marker":"[Bog+15]"},{"why":"Supplies Kato's inequality used to control the second derivative term in the energy dissipation.","marker":"[Kat72]"},{"why":"Provides the integration-by-parts statement for BV functions on the torus, used to drop spatial derivative terms.","marker":"[EG15]"}],"fun_headline_variants":["Cross-diffusion with independent drifts: existence proven","No self-diffusion? Totally mixed solutions exist","First existence result for mixed densities and differing drifts","BV regularity and weak solutions for cross-diffusion systems","Global weak solutions for drift-diffusion without self-diffusion"],"cache_read_input_tokens":30464,"weakest_assumption_plain":"The proof rests on an unproved transfer of an existing existence result for the regularised system from an interval with no-flux conditions to the circular domain; if that transfer fails, the regularised solutions on which every later estimate depends would not exist.","fun_headline_variants_meta":{"raw":{"variants":["Cross-diffusion with independent drifts: existence proven","No self-diffusion? Totally mixed solutions exist","First existence result for mixed densities and differing drifts","BV regularity and weak solutions for cross-diffusion systems","Global weak solutions for drift-diffusion without self-diffusion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000325,"raw_usage":{"total_tokens":1899,"prompt_tokens":1099,"completion_tokens":800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":715,"completion_tokens_details":{"reasoning_tokens":718}},"tokens_in":715,"tokens_out":800,"duration_ms":7444,"temperature":1.0,"reasoning_tokens":718,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:16:24.938601+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify the proof of Theorem 3.2 directly on the torus: show that the cited existence theorem for the η-regularised system remains valid on R/Z with the same energy and constants. If there is a pair of initial data satisfying H1-H4 and potentials in $W^{{2,1}}$(T) with V1−V2 in $W^{{3,1}}$(T) for which the regularised system has no weak solution on the torus, the approximation step collapses and the central claim is unsupported. Equivalently, a counterexample to Theorem 2.2 satisfying H1-H4 would refute the claim.","supporting_citations":[],"review_version":1}