{"id":"974a852c-e87c-4ea6-80ea-eba89e2f89f4","arxiv_id":"2606.25781","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a 1D cross-diffusion system with independent drifts, the same segregated initial data admits both a segregated weak solution and a mixing vanishing-viscosity solution.","lead":"For a two-species cross-diffusion model, this paper proves that when external forces push the two populations together, whether their densities overlap or stay separated depends on which definition of 'solution' you pick. It shows the same starting configuration can have two different weak solutions, so the model is not well-posed.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-uniqueness claim hinges on an unproved existence/compactness assumption: VV3 convergence of q_n is asserted via [30,48,51,58] but not verified for the segregated initial data.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap: existence of a vanishing-viscosity solution satisfying VV3 for the segregated initial data. The internal mathematics of the entropy inequality and the explicit example are sound; the contradiction is airtight once VV3 is granted. No internal inconsistency or sign error was found in the proofs of Propositions 3.1–3.2, Theorem 1.6, Lemma 3.3, or the explicit contradiction in Theorem 1.7. The only serious weakness is the reliance on cited preprints [30,48,51,58] for the existence of a vanishing-viscosity solution, and on [56] for the structure of segregated solutions, without stating or verifying the precise hypotheses and compactness results. This is a standard reason for a conditional acceptance rather than a rejection: the paper proves a conditional theorem of independent interest, and the missing piece is an external verification. A concrete check would settle it by inspecting the cited existence proofs for the VV3 convergence. Therefore the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":21683,"tokens_out":13194,"duration_ms":138673,"concrete_test":"Check whether the proof in [58] (or [30,48,51]) actually yields (VV3) for the explicit data (1.10) with f′(s)=logs and V,W satisfying (A2)–(A3): trace the compactness argument to see if q_n=∂xρ_n−(ρ_n/S_n)∂xS_n is shown to converge in L^1(0,T;H^{-m}_loc). If the cited proof only establishes weak L^1 convergence of ρ_n and μ_n (VV2) and not of q_n, then Definition 1.3 is not satisfied and the non-uniqueness claim is unproved. A minimal decisive check is to ask the authors to provide the precise theorem from [58] and the proof of the VV3 bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 3.2 is internally sound given (VV3): concavity of h gives upper semicontinuity of Φ and the pairing with ∂xV−∂xW passes to the limit via weak convergence in L^1(0,T;H^{-m}_loc). The contradiction for the explicit stationary solution in Theorem 1.7 is also correct: if (ρ̄,μ̄) were a vanishing-viscosity solution, (VV3) would force q=−σ*δ_0, yielding Φ≥tσ*(∂xW(0)−∂xV(0))>0 while Φ≡0. However, the paper never proves that a vanishing-viscosity solution actually exists for the initial datum (1.10). Remark 1.4 cites [30,48,51,58] for existence, but no theorem statement, hypotheses, or verification that those constructions produce (VV1)–(VV3) is given. In particular, (VV3) requires compactness of q_n=∂xρ_n−(ρ_n/S_n)∂xS_n in L^1(0,T;H^{-m}_loc), a nontrivial estimate not obviously implied by the cited existence proofs. If (VV3) fails for those sequences, Inequality (1.14) does not apply and Corollary 1.8 collapses. The same gap affects Theorem 1.9, which additionally assumes the segregated solution of [56] can be identified with the vanishing-viscosity solution of [58]; this compatibility is not stated. Thus the paper establishes a conditional non-uniqueness: the condition is not discharged.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a one-dimensional cross-diffusion system with common pressure but independent external potentials. It constructs an explicit stationary segregated solution for specially prepared initial data (Theorem 1.6) and proves that every vanishing-viscosity solution in the sense of Definition 1.3 satisfies the relative-entropy inequality (1.14) (Theorem 1.7). For the explicit segregated state the distributional quantity q equals -sigma_* delta_0, so (1.14) forces strictly positive relative entropy, contradicting segregation. This yields non-uniqueness of weak solutions under the assumption that at least one vanishing-viscosity solution exists (Corollary 1.8). The final section extends the argument to general segregated data under assumptions (B1)-(B4), using the structure and regularity of segregated solutions from [56] to prove that the right-hand side of (1.14) is positive on a time interval of positive measure, giving Theorem 1.9.","tokens_in":22049,"tokens_out":5947,"duration_ms":60868,"significance":"The relative-entropy mechanism developed here is original and potentially influential: it identifies a specific functional whose dissipation is controlled by the drift difference and whose weak upper semicontinuity passes through the viscous limit. The explicit stationary solution is clean, and the contradiction argument for that solution is formally correct. If the missing existence/compactness point is supplied, the result would be a notable non-uniqueness theorem for a degenerate cross-diffusion system, complementing the recent Muskat-type examples. However, as written, the main theorem is conditional on the existence of a vanishing-viscosity solution satisfying (VV3), which is asserted but not demonstrated.","major_comments":[{"comment":"The existence of a vanishing-viscosity solution satisfying (VV3) is not proved. Remark 1.4 refers to [30,48,51,58], but no theorem statement or verification is given that those constructions yield weak convergence of q_n = ∂_x ρ_n − (ρ_n/S_n)∂_x S_n in L^1(0,T;H^{-m}_loc). This is load-bearing: Proposition 3.2 and Inequality (1.14) are stated only for solutions in this class, and Corollary 1.8 requires at least one such solution. If (VV3) fails, the contradiction with the explicit segregated state collapses. The authors should state a precise existence theorem covering the data and pressure ranges used, and prove or cite a result that delivers (VV3).","section":"Definition 1.3, Remark 1.4"},{"comment":"The proof assumes that the segregated solution of [56] 'can be obtained from a vanishing viscosity approximation by the existence theory of [58, Theorem 1.3].' This is an unproved compatibility statement. To establish non-uniqueness it is enough to have two distinct solutions, one segregated and one vanishing-viscosity; the contradiction argument requires the [58] solution to be a vanishing-viscosity solution in the sense of Definition 1.3 for the same data, including the compactness of q_n. The paper does not verify (VV1)-(VV3) for the sequence constructed in [58]. Without this, Theorem 1.9 remains conditional.","section":"Theorem 1.9"},{"comment":"The passage 'Since Φ is concave, it is upper semi-continuous for the weak L1 convergence' is used to pass from limsup_n Φ(ρ_n(t), μ_n(t)) to Φ(ρ(t), μ(t)). This should be justified by a proof or a precise reference. The issue is not merely cosmetic: the weak convergence in (VV2) is only for a.e. t, and h is not globally bounded, so the standard semicontinuity result for concave integrals should be stated explicitly. The proof also drops the initial entropy term by positivity; this is fine, but the semicontinuity step needs to be made rigorous.","section":"Proposition 3.2"}],"minor_comments":[{"comment":"The phrase 'We can take the derivative ∂x on each side of the line' should read 'on each half-line' for clarity.","section":"Theorem 1.6 proof"},{"comment":"The two weak equations are displayed consecutively without a separator; adding a blank line or numbering the equations would improve readability.","section":"Definition 1.2"},{"comment":"The text cites several works as '2025', '2026', or 'In preparation' (e.g., [13,30,48,51,56,58]). For a journal submission, full bibliographic data or a statement of availability would be helpful, especially because the main proof depends on results from those works.","section":"Section 1.2"},{"comment":"In the slow-diffusion subsolution construction, the constants A, B, ε, c_λ are chosen in a nested way. It would help to list the order of choices explicitly: ε, then A/B, then A, then B, then c_λ, then τ̂. This would make it easier to verify that no circular dependence occurs.","section":"Proposition 4.11"},{"comment":"In Proposition 4.13, the notation ∂_x ρ_t is slightly informal; since ρ_t is a spatial density at fixed t, writing ∂_x ρ(t,·) would be clearer.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central claim is interesting but the main gap—existence of vanishing-viscosity solutions with the specific compactness (VV3)—is heavily dependent on several unpublished or very recent works. The editor may wish to verify that those references are available in a form that actually supports the asserted compactness, since the paper does not spell out the relevant theorems. The present version reads as a mathematically sound conditional result whose hypothesis is not discharged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe headline: this paper shows non-uniqueness for the Busenberg–Travis cross-diffusion system with independent drifts, and the core entropy mechanism is right. The caveat is that the result is conditional on a compactness property (VV3) for vanishing-viscosity solutions that is asserted by reference, not proved.\n\nWhat's new and good: the relative entropy Φ = ∫ h(ρ, μ) is well chosen; its dissipation at the viscous level (Prop 3.1) is an exact identity, and the lower bound (1.14) follows cleanly by weak convergence once VV3 holds. The explicit stationary segregated solution in Theorem 1.6 is simple and does its job: plugging it into the lower bound gives a strictly positive right-hand side while Φ≡0, so it cannot be the vanishing-viscosity limit. That is a real contradiction, not a formal one. The method is original, and the paper is honest about the independent parallel work [13].\n\nThe soft spot: the theorem as stated is conditional. Corollary 1.8 assumes existence of at least one vanishing-viscosity solution satisfying VV3. The authors say this follows from [30,48,51,58], but they do not state the relevant theorem or verify that those existence proofs produce weak convergence of q_n = ∂xρ_n − (ρ_n/S_n)∂xS_n in L^1(0,T;H^{-m}_{loc}). That is a genuine gap: without VV3, the entropy inequality does not apply. It may be that the cited papers do imply it, but the burden is on the authors to show how. The same issue affects Theorem 1.9, which additionally assumes that the segregated solution from [56] can be identified with the vanishing-viscosity solution from [58]; that identification is asserted, not proved. So the paper proves conditional non-uniqueness, with a condition the reader cannot discharge without going to the preprints.\n\nOn balance, the argument is internally sound conditional on that compactness; nothing in the logic looks broken. The paper deserves a serious referee. I would ask the authors to either prove VV3 for their data or state a precise lemma with the exact hypotheses from the cited papers, and to tighten the compatibility step in Theorem 1.9. If that is done, the result stands. For PDE analysts working on cross-diffusion, free boundaries, and non-uniqueness, this is a worthwhile read.","headline":"Entropy inequality is clean and the explicit counterexample works; the non-uniqueness is conditional on a compactness claim that is outsourced to references.","tokens_in":22514,"tokens_out":2943,"would_cite":true,"duration_ms":31436,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","35K65","35A02","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a two-species cross-diffusion system with independent drifts admits at least two distinct weak solutions for the same initial data, so weak well-posedness fails.","keywords":["cross-diffusion","non-uniqueness","weak solutions","vanishing viscosity","segregated solutions","relative entropy","free boundaries","drift-diffusion"],"falsifier":"Compute or construct the vanishing-viscosity limit for the explicit initial datum (1.10) under the condition ∂xW(0)−∂xV(0)>0. If the limiting densities remain segregated (Φ ≡ 0) for positive time, Inequality (1.14) would fail, and the theorem would be false. Conversely, any numerical or analytical demonstration of a positive overlap for this limit would support the claim.","tokens_in":21572,"feed_emoji":"🔄","tokens_out":5753,"duration_ms":54218,"temperature":0.7,"pith_summary":"This paper studies a one-dimensional system of two populations whose densities diffuse through a common pressure that depends on the total density, while each population is advected by its own external potential. Starting from initially segregated densities with a single interface, the authors show that the evolution is not fixed by the equations alone: the choice of solution concept changes the outcome. When the external potentials push the two phases toward each other at the interface, the vanishing-viscosity limit creates an overlap of the densities, while a segregated weak solution also exists. These two solutions are distinct, so the Cauchy problem is not well-posed in the class of weak solutions. The proof rests on a relative-entropy inequality that every vanishing-viscosity solution must satisfy, which the segregated solution violates.","feed_headline":"Cross-diffusion with independent drifts admits two weak solutions","feed_subtitle":"Vanishing-viscosity and segregated solution concepts disagree when drifts push phases together.","key_machinery":"The key object is the relative entropy Φ[ρ,µ] = ∫_R h(ρ,µ) dx with h(u,v) = (u+v)log(u+v)−u log u − v log v, which vanishes exactly in segregated states and measures the overlap. The main identity is the entropy dissipation bound at the viscous level: d/dt Φ ≥ ∫ (∂xV−∂xW) q dx, and passing to the limit gives the lower bound (1.14). The load-bearing identification is Lemma 3.3, which shows that for a segregated density S with a single interface at η, the distribution q = ∂xρ − (ρ/S)∂xS equals −S(η) δ_η. This turns the drift difference into a strictly positive contribution when the drifts converge at the interface.","core_discovery":"The central claim is that the Cauchy problem for the cross-diffusion system (1.1)–(1.2) admits at least two distinct weak solutions for the same segregated initial data, provided the drifts satisfy ∂xW(0)−∂xV(0)>0. The segregated solution is an explicit stationary state whose densities meet at the interface with a common value; it is a weak solution by direct construction. The authors prove a lower bound on the relative entropy Φ[ρ(t),µ(t)] for any vanishing-viscosity solution, namely Φ ≥ ∫_0^t ⟨q, ∂xV−∂xW⟩ ds, where q is the distributional limit of q_n = ∂xρ_n − (ρ_n/S_n)∂xS_n. For a segregated state with a single interface, q equals −S(0) δ_0, so the right-hand side is strictly positive, w","pith_inferences":["This suggests that any well-posedness theory for such cross-diffusion systems must either restrict the class of admissible weak solutions or impose an additional selection criterion, such as entropy or vanishing-viscosity admissibility.","The relative-entropy inequality could be tested numerically: computing the vanishing-viscosity limit for the explicit initial data should show a positive overlap, whereas the segregated stationary solution has zero overlap; observing otherwise would challenge the theorem.","The mechanism may extend to higher dimensions or to systems with more than two species, though the paper only treats one dimension; if the interface instability persists there, the notion of solution would need to be supplemented by physical selection principles."],"forward_implications":["The Cauchy problem for the system (1.1)–(1.2) is not well-posed in the class of weak solutions: selecting a solution by vanishing viscosity or by segregation yields different evolutions.","The interface behavior depends on the approximation scheme; sharp interfaces break down spontaneously under the vanishing-viscosity limit when the drifts force the phases together.","The relative entropy inequality (1.14) provides a necessary condition that any vanishing-viscosity solution must satisfy, ruling out segregated states in this regime.","For power-type pressures with α>1/3 and general segregated data, the non-uniqueness persists, so the phenomenon is not specific to the explicit stationary profile."],"fun_headline_variants":["Two weak solutions for cross-diffusion with independent drifts","Non-unique weak solutions when drifts push phases together","Cross-diffusion: independent drifts yield multiple weak solutions","Vanishing-viscosity and segregated solutions differ in cross-diffusion","Cauchy problem not well-posed for cross-diffusion with independent drifts"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on the unproven assertion that a vanishing-viscosity solution exists for the segregated initial data with the weak compactness specified in condition (VV3); the paper cites prior work for this, and without it the core inequality cannot be applied.","fun_headline_variants_meta":{"raw":{"variants":["Two weak solutions for cross-diffusion with independent drifts","Non-unique weak solutions when drifts push phases together","Cross-diffusion: independent drifts yield multiple weak solutions","Vanishing-viscosity and segregated solutions differ in cross-diffusion","Cauchy problem not well-posed for cross-diffusion with independent drifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1135,"prompt_tokens":703,"completion_tokens":432,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":341}},"tokens_in":447,"tokens_out":432,"duration_ms":4380,"temperature":1.0,"reasoning_tokens":341,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T10:09:36.169181+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute or construct the vanishing-viscosity limit for the explicit initial datum (1.10) under the condition ∂xW(0)−∂xV(0)>0. If the limiting densities remain segregated (Φ ≡ 0) for positive time, Inequality (1.14) would fail, and the theorem would be false. Conversely, any numerical or analytical demonstration of a positive overlap for this limit would support the claim.","supporting_citations":[],"review_version":2}