Global existence for a system without self-diffusion and different mobilities

Charles Elbar

arxiv: 2604.14775 · v1 · submitted 2026-04-16 · 🧮 math.AP

Global existence for a system without self-diffusion and different mobilities

Charles Elbar This is my paper

Pith reviewed 2026-05-10 10:35 UTC · model grok-4.3

classification 🧮 math.AP

keywords cross-diffusionweak solutionsglobal existenceapproximation schemesYoung measuresentropy estimatesone dimensiontorus

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The pith

Any admissible approximation sequence converges to a global weak solution of the one-dimensional cross-diffusion system with different mobilities.

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

The paper proves that for a cross-diffusion system of two populations in one dimension on the torus, with no self-diffusion, different mobilities, and linear pressure, global weak solutions exist for any bounded non-negative initial data. This is achieved by showing that any sequence of approximations satisfying certain admissibility conditions has a subsequence that converges to a solution of the system. The proof uses entropy estimates to gain compactness and applies the div-curl lemma within the theory of Young measures to handle the nonlinear terms. A reader would care because this provides a robust way to establish existence even when standard parabolic regularity is unavailable due to the absence of self-diffusion and unequal mobilities.

Core claim

For arbitrary bounded non-negative initial data, any admissible approximation sequence admits a subsequence converging to a weak solution of the cross-diffusion system. The system is one-dimensional on the torus with linear pressure law and different mobilities for the two species.

What carries the argument

An admissible approximation sequence, which is a sequence of approximate solutions satisfying specific conditions that allow passage to the limit using entropy bounds and Young measure techniques.

Load-bearing premise

The approximation sequence must meet the admissibility conditions, the pressure law is linear, and the setting is restricted to one spatial dimension on the torus.

What would settle it

An explicit construction of an admissible approximation sequence whose limit fails to satisfy the weak form of the cross-diffusion equations, or a counterexample showing non-existence of global weak solutions for some bounded initial data.

Figures

Figures reproduced from arXiv: 2604.14775 by Charles Elbar.

read the original abstract

We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a linear pressure law and different mobilities. For arbitrary bounded non-negative initial data, we show that any good approximation scheme, yields existence of global weak solutions. More precisely, we introduce a notion of \textit{admissible approximation sequence} and show that any such sequence admits a subsequence converging to a weak solution of the system. The strategy relies on entropy estimates and the div--curl lemma, in the framework of Young measures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper shows conditional global weak solutions exist for a 1D cross-diffusion system with unequal mobilities and linear pressure by verifying that admissible approximations converge via entropy estimates and the div-curl lemma.

read the letter

The core result is that any admissible approximation sequence for this two-population system on the torus converges (in the Young-measure sense) to a global weak solution, provided the pressure is linear. This extends earlier existence theorems to the case of different mobilities without a self-diffusion term, which had been a sticking point in some prior work on cross-diffusion models. The strategy is straightforward: entropy dissipation gives the necessary L1 and integrability bounds, then the div-curl lemma handles the passage to the limit in the flux terms. That part looks clean and uses standard tools without obvious circularity. The restrictions to one dimension, linear pressure, and admissible schemes are stated up front, so the claim does not overreach. The main limitation is that the result is conditional on the approximations satisfying the paper's entropy and integrability conditions; it does not construct such a sequence from arbitrary bounded initial data or extend immediately to higher dimensions or nonlinear pressure. Without the full details of how the approximations are built, it is hard to judge how broadly applicable the theorem will be in practice. This is niche work aimed at people already working on cross-diffusion systems and compensated-compactness methods. A reader in that subfield will see a modest but technically solid increment. I would send it to peer review; the argument is coherent on its own terms and fills a specific gap even if the scope stays narrow.

Referee Report

2 major / 3 minor

Summary. The manuscript studies a one-dimensional cross-diffusion system on the torus for two populations with different mobilities, no self-diffusion, and linear pressure law. For arbitrary bounded non-negative initial data, the authors introduce the notion of an admissible approximation sequence and prove that any such sequence admits a subsequence converging in the sense of Young measures to a global weak solution. The argument relies on entropy dissipation estimates to obtain integrability bounds followed by an application of the div-curl lemma to pass to the limit in the flux terms.

Significance. If the central compactness argument closes rigorously, the result is of moderate significance: it supplies a conditional existence theorem for a degenerate cross-diffusion system in which the usual self-diffusion regularisation is absent. The restriction to one dimension and linear pressure allows the compensated-compactness step to succeed, and the conditional formulation (existence along admissible schemes) is stated explicitly. This approach may serve as a template for other 1D degenerate parabolic systems where direct construction of solutions is difficult.

major comments (2)

[§3] §3 (entropy estimates): the dissipation identity obtained from the linear pressure must be shown to produce uniform L^1 bounds independent of the approximation parameter; the paper should verify that the different mobilities do not destroy the coercivity needed for the subsequent Young-measure compactness.
[§5] §5 (div-curl application): the identification of the limit flux via the div-curl lemma in Young measures relies on the specific algebraic structure induced by the linear pressure; a short remark explaining why the same argument fails for nonlinear pressures would clarify the scope of the method.

minor comments (3)

[Abstract] The abstract uses the informal phrase 'any good approximation scheme'; the precise definition of 'admissible' appears only in §2. Aligning the terminology in the abstract with the later definition would improve readability.
[Theorem 1.1] Theorem 1.1 states convergence in the sense of Young measures; the precise topology (e.g., narrow convergence of measures or weak-* convergence in L^∞) should be recalled explicitly in the statement.
[Introduction] A brief comparison with existing unconditional existence results that require self-diffusion or equal mobilities would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address the two major comments point by point below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (entropy estimates): the dissipation identity obtained from the linear pressure must be shown to produce uniform L^1 bounds independent of the approximation parameter; the paper should verify that the different mobilities do not destroy the coercivity needed for the subsequent Young-measure compactness.

Authors: In Section 3 we obtain the entropy dissipation identity (3.5) from the linear pressure, which directly controls ∫ |∇√u_i|^2 dx uniformly in the approximation parameter. The different mobilities appear as positive prefactors in the resulting quadratic form; because the cross terms cancel by the entropy structure, coercivity is preserved for any m1, m2 > 0. The uniform L^1 bounds then follow from the 1D Sobolev embedding. To make the argument fully explicit we will add a short clarifying paragraph after (3.5) and a remark confirming independence of the mobilities. revision: yes
Referee: [§5] §5 (div-curl application): the identification of the limit flux via the div-curl lemma in Young measures relies on the specific algebraic structure induced by the linear pressure; a short remark explaining why the same argument fails for nonlinear pressures would clarify the scope of the method.

Authors: We agree that the linear pressure is essential: it produces the precise algebraic cancellation (identity (5.7)) that allows the div-curl lemma to identify the limit flux in the Young-measure sense. For a nonlinear pressure the flux terms generally fail to satisfy the required orthogonality condition, so the same passage to the limit does not hold. We will insert a brief remark at the end of Section 5 explaining this limitation and the consequent restriction of the method to the linear-pressure case. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes global weak solutions for a 1D cross-diffusion system by showing that any admissible approximation sequence (defined explicitly via entropy and integrability conditions) has a subsequence converging in the Young-measure sense to a weak solution. The argument relies on standard entropy dissipation for L1 bounds and the div-curl lemma to pass to the limit in the flux terms. These are external, independently verifiable mathematical tools with no reduction to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The restrictions (linear pressure, 1D torus, admissible schemes) are stated explicitly as necessary for the compactness step and do not create circularity. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard analytic tools and the definition of admissible approximations; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math The div-curl lemma and properties of Young measures apply in the compensated compactness framework for passing to the limit.
Invoked explicitly in the strategy for convergence of the approximation sequence.
domain assumption The domain is the one-dimensional torus with periodic boundary conditions.
The system is posed on the torus as stated in the abstract.

pith-pipeline@v0.9.0 · 5366 in / 1251 out tokens · 38280 ms · 2026-05-10T10:35:30.245750+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

[1]J. M. Ball,A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transitions, M. Rascle, D. Serre, and M. Slemrod, eds., vol. 344 of Lecture Notes in Physics, Springer, Berlin, Heidelberg, 1989, pp. 207–215. [2]M. Bertsch, M. E. Gurtin, and D. Hilhorst,On interacting populations that disperse to avoid crowding: ...

work page 1989
[2]

Bertsch, M

[3]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. [4]S. N. Busenberg and C. C. Tra vis,Epidemic models with spatial spread due to popula- tion migration, Journal of Mathematical Biology, 16 (1983), pp. 181–198. [5]J. A...

work page 1985
[3]

[6]L. Chen, E. S. Daus, A. Holzinger, and A. Jüngel,Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems, Journal of Nonlinear Science, 31 (2021), p. Art

work page 2021
[4]

[7]L. Chen, E. S. Daus, and A. Jüngel,Rigorous mean-field limit and cross diffusion, Zeitschrift für angewandte Mathematik und Physik, 70 (2019), p. Art

work page 2019
[5]

Conti, G

[8]S. Conti, G. Dolzmann, and S. Müller,The div-curl lemma for sequences whose divergence and curl are compact inW−1,1, C. R. Math. Acad. Sci. Paris, 349 (2011), pp. 175–

work page 2011
[6]

Elbar and F

[9]N. Da vid, T. Debiec, M. Mandal, and M. Schmidtchen,A degenerate cross-diffusion system as the inviscid limit of a nonlocal tissue growth model, SIAM J. Math. Anal., 56 (2024), pp. 2090–2114. 16 [10]M. Doumic, S. Hecht, B. Perthame, and D. Peurichard,Multispecies cross- diffusions: From a nonlocal mean-field to a porous medium system without self-diffu...

work page arXiv 2024
[7]

[15]M. E. Gurtin and A. C. Pipkin,A note on interacting populations that disperse to avoid crowding, Quarterly of Applied Mathematics, 42 (1984), pp. 87–94. [16]P. Gwiazda, B. Perthame, and A. Świerczewska-Gwiazda,A two-species hyperbolic-parabolic model of tissue growth, Communications in Partial Differential Equa- tions, 44 (2019), pp. 1605–1618. [17]K....

work page arXiv 1984

[1] [1]

[1]J. M. Ball,A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transitions, M. Rascle, D. Serre, and M. Slemrod, eds., vol. 344 of Lecture Notes in Physics, Springer, Berlin, Heidelberg, 1989, pp. 207–215. [2]M. Bertsch, M. E. Gurtin, and D. Hilhorst,On interacting populations that disperse to avoid crowding: ...

work page 1989

[2] [2]

Bertsch, M

[3]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. [4]S. N. Busenberg and C. C. Tra vis,Epidemic models with spatial spread due to popula- tion migration, Journal of Mathematical Biology, 16 (1983), pp. 181–198. [5]J. A...

work page 1985

[3] [3]

[6]L. Chen, E. S. Daus, A. Holzinger, and A. Jüngel,Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems, Journal of Nonlinear Science, 31 (2021), p. Art

work page 2021

[4] [4]

[7]L. Chen, E. S. Daus, and A. Jüngel,Rigorous mean-field limit and cross diffusion, Zeitschrift für angewandte Mathematik und Physik, 70 (2019), p. Art

work page 2019

[5] [5]

Conti, G

[8]S. Conti, G. Dolzmann, and S. Müller,The div-curl lemma for sequences whose divergence and curl are compact inW−1,1, C. R. Math. Acad. Sci. Paris, 349 (2011), pp. 175–

work page 2011

[6] [6]

Elbar and F

[9]N. Da vid, T. Debiec, M. Mandal, and M. Schmidtchen,A degenerate cross-diffusion system as the inviscid limit of a nonlocal tissue growth model, SIAM J. Math. Anal., 56 (2024), pp. 2090–2114. 16 [10]M. Doumic, S. Hecht, B. Perthame, and D. Peurichard,Multispecies cross- diffusions: From a nonlocal mean-field to a porous medium system without self-diffu...

work page arXiv 2024

[7] [7]

[15]M. E. Gurtin and A. C. Pipkin,A note on interacting populations that disperse to avoid crowding, Quarterly of Applied Mathematics, 42 (1984), pp. 87–94. [16]P. Gwiazda, B. Perthame, and A. Świerczewska-Gwiazda,A two-species hyperbolic-parabolic model of tissue growth, Communications in Partial Differential Equa- tions, 44 (2019), pp. 1605–1618. [17]K....

work page arXiv 1984