Interfaces and non-uniqueness in a cross-diffusion system with independent drifts

Charles Elbar; Guy Parker

REVIEW 3 major objections 5 minor 16 references

Interfaces and non-uniqueness in a cross-diffusion system with independent drifts

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Entropy inequality is clean and the explicit counterexample works; the non-uniqueness is conditional on a compactness claim that is outsourced to references. read the letter →

arxiv 2606.25781 v2 pith:VYFMK556 submitted 2026-06-24 math.AP

Charles Elbar , Guy Parker This is my paper · ORCID

classification math.AP MSC 35K5535K6535A0235D30

keywords cross-diffusionnon-uniquenessweaksolutionsvanishingviscositysegregatedrelativeentropyfreeboundariesdrift-diffusion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear —,

The 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.

What'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].

The 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.

On 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Definition 1.3, Remark 1.4] 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).
[Theorem 1.9] 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.
[Proposition 3.2] 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.

minor comments (5)

[Theorem 1.6 proof] The phrase 'We can take the derivative ∂x on each side of the line' should read 'on each half-line' for clarity.
[Definition 1.2] The two weak equations are displayed consecutively without a separator; adding a blank line or numbering the equations would improve readability.
[Section 1.2] 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.
[Proposition 4.11] 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.
[Notation] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the relative-entropy inequality is derived from the viscous equations and the stated compactness (VV3); existence of a vanishing-viscosity solution is invoked as an external theorem, not built into the derivation.

full rationale

The core derivation is self-contained and non-circular. Theorem 1.6 explicitly constructs a stationary segregated weak solution. Proposition 3.1 computes the exact dissipation of the relative entropy for smooth solutions of the viscous system, giving d/dt Φ ≥ ∫(∂xV−∂xW)qε dx. Proposition 3.2 passes to the limit using concavity of h (Proposition 1.5), weak-L^1 convergence of the densities, and the compactness/continuity condition (VV3) built into Definition 1.3. Theorem 1.7 then applies Lemma 3.3 to the explicit segregated stationary solution: if it were a vanishing-viscosity solution, (VV3) would force q = −σ∗δ0, making the right-hand side of the inequality strictly positive while Φ ≡ 0, a contradiction. Each step follows from the paper's own equations and definitions. There are no fitted parameters, no quantity is defined in terms of the target conclusion, and no load-bearing argument reduces to a self-citation. The paper explicitly conditions Corollary 1.8 on the existence of at least one vanishing-viscosity solution, and Remark 1.4 explicitly states that the existence of such solutions is a separate theorem proved in [30,48,51,58]. The proof of Theorem 1.9 similarly relies on the external existence result [58, Theorem 1.3] and on the segregated-solution structure from [56]. Whether those cited results actually supply the precise compactness (VV3) for the relevant initial data is a genuine, and potentially serious, correctness/existence gap, but it is a missing verification, not a circular reduction. The manuscript itself flags that this compactness is an input to the method rather than obtained within the paper. Under the rule that absence of verification is not circularity, the appropriate finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof is a PDE analysis relying on cited existence/regularity theorems; no new physical entities or fitted constants. The only ad hoc element is the definition of a vanishing-viscosity solution requiring q_n compactness in (VV3), which is a definitional choice rather than an unstated axiom.

assumptions (6)

domain assumption Assumptions (A1)-(A3): f strictly convex with inverse g; V,W ∈ C^3, ∂xV,∂xW ∈ L∞, ∂xV−∂xW ∈ C_c^∞, and (A3) admissibility of the explicit profile.
These define the class where Theorems 1.6 and 1.7 are proved; they are hypotheses, not derived results.
domain assumption Assumptions (B1)-(B4): segregated initial data in L1∩L∞ with positivity near the interface, and pressure f_α with α > 1/3.
These restrict the general-data non-uniqueness theorem to power-type pressures and well-prepared data.
domain assumption Existence of vanishing-viscosity solutions satisfying Definition 1.3(VV3) for the relevant data is imported from [30,48,51,58].
Load-bearing: the paper does not construct these solutions itself; if VV3 is not satisfied, the entropy lower bound does not apply.
domain assumption Structural properties of segregated solutions (u_t monotone, H^1 in time, BV, continuity of S) are imported from [56, Theorem 3.5].
The general-data proof uses the representation of the interface and continuity of the aggregate density from this cited work.
standard math Comparison principle for pressure sub-solutions [42, Theorem 3.4] applies to the piecewise quadratic/travelling-wave sub-solutions constructed in Proposition 4.11.
The slow-diffusion persistence argument rests on this comparison principle; its hypotheses are not re-verified in the paper.
standard math Harnack inequality [25, Corollary 16.1] and the strong minimum principle [31] for the decoupled drift-diffusion equations in Propositions 4.6-4.7.
These are used to show that the decoupled densities are positive in the linear and fast-diffusion regimes.

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Pith. "Pith review of Interfaces and non-uniqueness in a cross-diffusion system with independent drifts." pith.science (2026). https://pith.science/paper/VYFMK556

@misc{pith2026260625781,
  author       = {Pith},
  title        = {Pith review of: Interfaces and non-uniqueness in a cross-diffusion system with independent drifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYFMK556}},
  note         = {Machine review of arXiv:2606.25781}
}

read the original abstract

We study a one-dimensional cross-diffusion system of two populations. Their densities are diffused with a common pressure that depends on the total density, but are transported by two independent external potentials. Starting from segregated initial data with a single interface, we show that the way the two densities meet is not given by the equation alone: it depends on the notion of solution one chooses. When the drifts push the two phases towards each other at the interface, the vanishing-viscosity solution creates an overlap, whereas a segregated weak solution also exists. The two solutions are distinct, so the Cauchy problem is not well posed in the class of weak solutions. We first prove the result for an explicit stationary segregated solution, and then extend the non-uniqueness to general segregated data.

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Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Alasio, M

[1]L. Alasio, M. Bruna, S. Fagioli, and S. Schulz,Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts, Nonlinear Anal., 223 (2022), pp. Paper No. 113064,

2022

[2] [3]

,On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities, Nonlinear Anal., 11 (1987), pp. 493–499. [4]M. Bertsch, M. E. Gurtin, D. Hilhorst, and L. A. Peletier,On interacting popula- tions that disperse to avoid crowding: preservation of segregation, J. Math. Biol., 23 (1985), pp. 1–13. [5]M. Bertsch, D. Hilhorst...

1987

[3] [10]

Jacobs,Existence of solutions to reaction cross diffusion systems, SIAM J

[38]M. Jacobs,Existence of solutions to reaction cross diffusion systems, SIAM J. Math. Anal., 55 (2023), pp. 6991–7023. [39]A. Jüngel, C. Reisch, and S. Xhahysa,Multiphase cross-diffusion models for tissue structures: modeling, analysis, numerics,

2023

[4] [11]

Kim and A

[40]I. Kim and A. R. Mészáros,On nonlinear cross-diffusion systems: an optimal transport approach, Calc. Var. Partial Differential Equations, 57 (2018), pp. Paper No. 79,

2018

[5] [12]

Kim and J

[41]I. Kim and J. Tong,Interface dynamics in a two-phase tumor growth model, Interfaces Free Bound., 23 (2021), pp. 191–304. [42]I. Kim and Y. P. Zhang,Regularity properties of degenerate diffusion equations with drifts, SIAM J. Math. Anal., 50 (2018), pp. 4371–4406. [43]M. Laborde,On cross-diffusion systems for two populations subject to a common conges-...

2021

[6] [14]

Noisette and L

[52]F. Noisette and L. Székelyhidi, Jr.,Mixing solutions for the Muskat problem with variable speed, J. Evol. Equ., 21 (2021), pp. 3289–3312. [53]F. Otto,Evolution of microstructure in unstable porous media flow: a relaxational ap- proach, Comm. Pure Appl. Math., 52 (1999), pp. 873–915. [54]B. C. Price and X. Xu,Global existence theorem for a model govern...

2021

[7] [15]

Serrin,Pathological solutions of elliptic differential equations, Ann

[57]J. Serrin,Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 18 (1964), pp. 385–387. [58]J. Skrzeczkowski,Global solutions to cross-diffusion systems with independent advections in one dimension,

1964

[8] [16]

Stará and O

[59]J. Stará and O. John,Some (new) counterexamples of parabolic systems, Comment. Math. Univ. Carolin., 36 (1995), pp. 503–510. [60]M. Struwe,A counterexample in regularity theory for parabolic systems, Czechoslovak Math. J., 34(109) (1984), pp. 183–188. [61]Y. B. Zel’dovič and A. S. Kompaneec,On the theory of propagation of heat with the heat conductivi...

1995