Pith. sign in

REVIEW 3 minor 49 references

Non-uniqueness of weak solutions to cross-diffusion systems with advection

T0 review · 0 major / 3 minor · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cross-diffusion system with advection admits both segregated and mixing weak solutions from the same half-line initial data.

desk verdict The paper gives an explicit construction of non-uniqueness for a cross-diffusion-advection system on the line, with one segregated solution and infinitely many mixing ones from half-line initial data. read the letter →

arxiv 2606.25783 v1 pith:YUAS7N4N submitted 2026-06-24 math.AP

José A. Carrillo , Ansgar Jüngel , Jakub Skrzeczkowski , Yao Yao This is my paper · ORCID
classification math.AP
keywords cross-diffusionadvectionweaksolutionsnon-uniquenesssegregationmixingpopulationdynamicspressureexponents
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

The paper constructs two distinct weak solutions for a cross-diffusion system with advection describing two segregating population species. From initial densities supported exactly on the half-lines x ≤ 0 and x ≥ 0, one solution keeps the densities confined to their supports and completely segregated for all time, while the second class of solutions allows the densities to invade the opposite half-line after a finite time, producing mixing. The construction yields infinitely many mixing solutions and holds for the full range of pressure exponents, with quantitative estimates on the mixing process available for certain exponents. A sympathetic reader would care because the result supplies one of the first explicit demonstrations of non-uniqueness for this class of equations together with a concrete mixing phenomenon.

What carries the argument

Explicit construction of a segregated weak solution and infinitely many mixing weak solutions for the cross-diffusion-advection system with initial data supported on complementary half-lines.

What would settle it

A proof that every weak solution remains segregated for all time with these initial data, or a numerical approximation that stays strictly segregated without any invasion, would falsify the existence of mixing solutions.

Watch

Extended reading notes

Core claim

Starting from two initial densities supported on the half-lines x≤0 and x≥0, respectively, the cross-diffusion-advection system on the whole line admits a segregated weak solution that remains confined to the initial supports and stays completely segregated, as well as infinitely many mixing weak solutions in which the densities begin to invade the opposite half-line after a finite time.

Load-bearing premise

The explicit construction of both segregated and mixing weak solutions is possible only for this specific cross-diffusion-advection system on the whole line with initial data exactly supported on complementary half-lines.

Editorial extensions

If this is right

  • Weak solutions to the system are not unique.
  • Mixing can occur in finite time even though the initial data are completely segregated.
  • Infinitely many distinct mixing solutions exist.
  • The non-uniqueness result holds for every pressure exponent.
  • Quantitative estimates on the rate or extent of mixing are available for a certain range of exponents.

Reading between the lines

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

  • Additional selection principles such as entropy or viscosity conditions may be needed to restore uniqueness in applications.
  • The half-line support geometry is essential to the construction, so the same non-uniqueness may not appear for initial data with different supports.
  • Numerical schemes could converge to either the segregated or a mixing solution depending on regularization or discretization details.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript constructs explicit weak solutions to a cross-diffusion system with advection on the real line for initial data supported on complementary half-lines x≤0 and x≥0. It produces a segregated solution whose supports remain disjoint for all time and infinitely many mixing solutions whose supports overlap after a finite time; the construction is claimed to hold for the full range of pressure exponents, with quantitative estimates on the mixing process available for a sub-range of exponents.

Significance. If the explicit constructions are verified to satisfy the weak formulation, the result supplies one of the first concrete demonstrations of non-uniqueness for this class of equations together with an explicit mixing mechanism. The direct, parameter-free nature of the construction (no fitted parameters or auxiliary entropy conditions invoked to select one family) and its coverage of all pressure exponents constitute clear strengths.

minor comments (3)
  1. The introduction should state the precise weak formulation (including the sense in which the advection and cross-diffusion terms are integrated) before the construction begins, so that the subsequent verification steps can be checked against a single displayed definition.
  2. [§4] Figure captions and the text describing the interface motion should use consistent notation for the free boundary locations; currently the symbols for the left- and right-moving fronts appear to be interchanged in one paragraph of §4.
  3. [Theorem 5.3] The quantitative estimates in the range p>2 are stated only for the L^1 distance between supports; an explicit statement of the constant dependence on the initial data and on p would make the result easier to compare with related literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the explicit constructions demonstrating non-uniqueness, and the recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper's central result is an explicit construction of both a segregated weak solution (supports remain disjoint) and mixing weak solutions (supports overlap after finite time) for the same initial data supported on complementary half-lines. This is presented as a direct verification that both families satisfy the weak form of the system, with no reduction of the claimed non-uniqueness to a fitted parameter, self-definition, or load-bearing self-citation chain. The construction is stated to apply for the full range of pressure exponents and is externally falsifiable by checking the weak formulation on the given data; no ansatz is smuggled in via prior work, and no uniqueness theorem is invoked to force the result. The derivation chain is therefore self-contained against the paper's own equations and initial data.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based solely on the abstract, the paper relies on standard mathematical definitions of weak solutions to PDEs and the well-posedness framework for cross-diffusion systems; no free parameters, invented entities, or ad-hoc axioms are mentioned.

assumptions (1)
  • standard math Existence and well-posedness framework for weak solutions to cross-diffusion systems with advection
    The construction presupposes that weak solutions are defined and that the system admits such solutions under the given initial data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-uniqueness of weak solutions to cross-diffusion systems with advection." pith.science (2026). https://pith.science/paper/YUAS7N4N

@misc{pith2026260625783,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of weak solutions to cross-diffusion systems with advection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUAS7N4N}},
  note         = {Machine review of arXiv:2606.25783}
}
abstract

A cross-diffusion system with advection is considered on the whole line, describing the dynamics of two segregating population species. Starting from two initial densities supported on the half-lines $x\leq 0$ and $x\geq 0$, respectively, we construct two distinct solutions of the system: one pair of densities remains confined to their initial supports and stay completely segregated, while the second pair of densities begins to invade the opposite half-line after a finite time (mixing). To the best of our knowledge, this provides one of the first examples of non-uniqueness for this class of equations, together with an explicit demonstration of mixing phenomena. The construction produces infinitely many mixing solutions and applies throughout the full range of pressure exponents. For a certain range of exponents, we additionally obtain quantitative estimates on the mixing process.

Figures

Figures reproduced from arXiv: 2606.25783 by the authors.

Figure 1
Figure 1. A segregated stationary solution for α = 2 with initial conditions u 0 , v 0 such that ´ R u 0 (x) dx = ´ R v 0 (x) dx = 1 2 . Both species have a jump at x = 0. However, s = u + v has the unique continuous representative and s(t, 0) is its value at x = 0. Step 1: Estimate on ´ R (u(t, x) log u(t, x) + v(t, x) log v(t, x)) dx. We claim that ˆ R [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 9 canonical work pages

  1. [1]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara.Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

    2000

  2. [2]

    N.J.Armstrong, K.J.Painter, andJ.A.Sherratt.Acontinuumapproachtomodellingcell-celladhesion. J. Theoret. Biol., 243(1):98–113, 2006

    2006

  3. [3]

    Bailo, J

    R. Bailo, J. A. Carrillo, and D. Gómez-Castro. Aggregation-diffusion equations for collective behaviour in the sciences. InRecent developments in industrial and applied mathematics, volume 1 ofICIAM2023 Springer Ser., pages 177–200. Springer, Singapore, [2026]©2026

    2026

  4. [4]

    Bansaye, A

    V. Bansaye, A. Moussa, and F. Muñoz Hernández. Stability of a cross-diffusion system and approxi- mation by repulsive random walks: a duality approach.J. Eur. Math. Soc. (JEMS), 27(9):3889–3928, 2025

    2025

  5. [5]

    Bertsch, R

    M. Bertsch, R. Dal Passo, and M. Mimura. A free boundary problem arising in a simplified tumour growth model of contact inhibition.Interfaces Free Bound., 12(2):235–250, 2010

    2010

  6. [6]

    Bertsch, M

    M. Bertsch, M. E. Gurtin, and D. Hilhorst. On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities.Nonlinear Anal., 11(4):493–499, 1987

    1987

  7. [7]

    Bertsch, M

    M. Bertsch, M. E. Gurtin, D. Hilhorst, and L. A. Peletier. On interacting populations that disperse to avoid crowding: preservation of segregation.J. Math. Biol., 23(1):1–13, 1985

    1985

  8. [8]

    Bertsch, D

    M. Bertsch, D. Hilhorst, H. Izuhara, and M. Mimura. A nonlinear parabolic–hyperbolic system for contact inhibition of cell-growth.Differ. Eqs. Appl., 4:137–157, 2012. 30 J. A. CARRILLO, A. JÜNGEL, J. SKRZECZKOWSKI, AND Y. YAO

    2012

Show all 49 references
  1. [9]

    Brezis.Functional analysis, Sobolev spaces and partial differential equations

    H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

    2011

  2. [10]

    Bubba, B

    F. Bubba, B. Perthame, C. Pouchol, and M. Schmidtchen. Hele–Shaw limit for a system of two reaction- (cross-)diffusion equations for living tissues.Arch. Ration. Mech. Anal., 236(2):735–766, 2020

    2020

  3. [11]

    Busenberg and C

    S. Busenberg and C. Travis. Epidemic models with spatial spread due to population migration.J. Math. Biol., 16:181–198, 1983

    1983

  4. [12]

    J. A. Carrillo, X. Chen, B. Du, and A. Jüngel. Fluid relaxation approximation of the Busenberg–Travis cross-diffusion system.Commun. Math. Phys., 406:paper no. 151, 29 pages, 2025

    2025

  5. [13]

    J. A. Carrillo, C. Elbar, S. Fronzoni, and J. Skrzeczkowski. The nonlocal-to-local limit approximating quadratic porous medium equation: rate of convergence via evolutionary variational inequality in one dimension.Commun. Pure Appl. Anal., 31(0):230–244, 2026

    2026

  6. [14]

    J. A. Carrillo, S. Fagioli, F. Santambrogio, and M. Schmidtchen. Splitting schemes and segregation in reaction cross-diffusion systems.SIAM J. Math. Anal., 50(5):5695–5718, 2018

    2018

  7. [15]

    J. A. Carrillo and S. Guo. Interacting particle approximation of cross-diffusion systems.Nonlinearity, 39(2):Paper No. 025009, 21, 2026

    2026

  8. [16]

    J. A. Carrillo, S. Guo, and A. Holzinger. Propagation of chaos for multi-species moderately interacting particle systems up to Newtonian singularity.arXiv preprint arXiv:2501.03087, 2025

    2025

  9. [17]

    J. A. Carrillo, P. Gwiazda, and J. Skrzeczkowski. A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations.arXiv preprint arXiv:2603.25634, 2026

    2026

  10. [18]

    J. A. Carrillo, Y. Huang, and M. Schmidtchen. Zoology of a nonlocal cross-diffusion model for two species.SIAM J. Appl. Math., 78(2):1078–1104, 2018

    2018

  11. [19]

    J. A. Carrillo, H. Murakawa, M. Sato, H. Togashi, and O. Trush. A population dynamics model of cell- cell adhesion incorporating population pressure and density saturation.J. Theoret. Biol., 474:14–24, 2019

    2019

  12. [20]

    J. A. Carrillo, Y. Salmaniw, and J. Skrzeczkowski. Well-posedness of aggregation-diffusion systems with irregular kernels.arXiv preprint arXiv:2406.09227, to appear in Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2024

    2024

  13. [21]

    J. A. Carrillo, J. Skrzeczkowski, and J. Warnett. The Stein-log-Sobolev inequality and the exponen- tial rate of convergence for the continuous Stein variational gradient descent method.arXiv preprint arXiv:2412.10295, 2024

    2024

  14. [22]

    L. Chen, E. Daus, and A. Jüngel. Rigorous mean-field limit and cross diffusion.Z. Angew. Math. Phys., 70:no. 122, 21 pages, 2019

    2019

  15. [23]

    X. Chen, E. S. Daus, and A. Jüngel. Global existence analysis of cross-diffusion population systems for multiple species.Arch. Ration. Mech. Anal., 227(2):715–747, 2018. NON-UNIQUENESS OF WEAK SOLUTIONS TO CROSS-DIFFUSION SYSTEMS 31

    2018

  16. [24]

    Chen and A

    X. Chen and A. Jüngel. Weak–strong uniqueness of renormalized solutions to reaction–cross-diffusion systems.Math. Models Meth. Appl. Sci., 29:237–270, 2019

    2019

  17. [25]

    Dębiec, B

    T. Dębiec, B. Perthame, M. Schmidtchen, and N. Vauchelet. Incompressible limit for a two-species model with coupling through Brinkman’s law in any dimension.J. Math. Pures Appl. (9), 145:204–239, 2021

    2021

  18. [26]

    Druet, K

    P.-E. Druet, K. Hopf, and A. Jüngel. Hyperbolic–parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion.Commun. Partial Differ. Eqs., 48:863–894, 2023

    2023

  19. [27]

    Druet and A

    P.-E. Druet and A. Jüngel. Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient.SIAM J. Math. Anal., 52:2179–2197, 2020

    2020

  20. [28]

    C. Elbar. Global existence for a system without self-diffusion and different mobilities.arXiv preprint arXiv:2604.14775, 2026

    2026 arXiv

  21. [29]

    Elbar and G

    C. Elbar and G. Parker. Interfaces and non-uniqueness in a cross-diffusion system with independent drifts.In preparation, 2026

    2026

  22. [30]

    Elbar and F

    C. Elbar and F. Santambrogio. A cross-diffusion system with independent drifts and fast diffusion.arXiv preprint arXiv:2510.07937, 2025

    2025

  23. [31]

    Elbar and J

    C. Elbar and J. Skrzeczkowski. On the inviscid limit connecting Brinkman’s and Darcy’s models of tissue growth with nonlinear pressure.J. Math. Fluid Mech., 27(2):paper no. 28, 14, 2025

    2025

  24. [32]

    Falcó, R

    C. Falcó, R. E. Baker, and J. A. Carrillo. A nonlocal-to-local approach to aggregation-diffusion equa- tions.SIAM Rev., 67(2):353–372, 2025

    2025

  25. [33]

    Galiano, S

    G. Galiano, S. Shmarev, and J. Velasco. Existence and multiplicity of solutions to a cell-growth contact inhibition problem.Discrete Cont. Dyn. Sys., 35:1479–1501, 2015

    2015

  26. [34]

    Giri and H

    V. Giri and H. Kwon. On non-uniqueness of continuous entropy solutions to the isentropic compressible euler equations.Arch. Ration. Mech. Anal., 245(2):1213–1283, 2022

    2022

  27. [35]

    H. P. Greenspan. On the growth and stability of cell cultures and solid tumors.J. Theoret. Biol., 56(1):229–242, 1976

    1976

  28. [36]

    M. E. Gurtin and A. C. Pipkin. A note on interacting populations that disperse to avoid crowding. Quart. Appl. Math., 42(1):87–94, 1984

    1984

  29. [37]

    Gwiazda, B

    P. Gwiazda, B. Perthame, and A. Świerczewska Gwiazda. A two-species hyperbolic-parabolic model of tissue growth.Comm. Partial Differential Equations, 44(12):1605–1618, 2019

    2019

  30. [38]

    Hopf and A

    K. Hopf and A. Jüngel. Convergence of a finite volume scheme and dissipative measure-valued–strong stability for a hyperbolic–parabolic cross-diffusion system.Numer. Math., 157:951–992, 2025

    2025

  31. [39]

    M. Jacobs. Lagrangian solutions to the porous media equation and reaction diffusion systems.SIAM J. Math. Anal., 57(4):3806–3839, 2025

    2025

  32. [40]

    Kim and A

    I. Kim and A. R. Mészáros. On nonlinear cross-diffusion systems: an optimal transport approach.Calc. Var. Partial Differential Equations, 57(3):Paper No. 79, 40, 2018. 32 J. A. CARRILLO, A. JÜNGEL, J. SKRZECZKOWSKI, AND Y. YAO

    2018

  33. [41]

    Laurençot and B.-V

    P. Laurençot and B.-V. Matioc. Weak-strong uniqueness for a class of degenerate parabolic cross- diffusion systems.Arch. Math. (Brno), 59(2):201–213, 2023

    2023

  34. [42]

    Matthes and C

    D. Matthes and C. Parsch. Convergence to equilibrium for cross diffusion systems with nonlocal inter- action.Calc. Var. Partial Differential Equations, 65(3):Paper No. 74, 2026

    2026

  35. [43]

    A. R. Mészáros and G. Parker. Existence theory for a cross-diffusion system with independent drifts: Mixing dynamics.arXiv preprint arXiv:2603.18770, 2026

    2026

  36. [44]

    Murakawa and H

    H. Murakawa and H. Togashi. Continuous models for cell–cell adhesion.Journal of Theoretical Biology, 374:1–12, 2015

    2015

  37. [45]

    Perthame, F

    B. Perthame, F. Quirós, and J. L. Vázquez. The Hele–Shaw asymptotics for mechanical models of tumor growth.Arch. Ration. Mech. Anal., 212(1):93–127, 2014

    2014

  38. [46]

    B. C. Price and X. Xu. Global existence theorem for a model governing the motion of two cell popula- tions.Kinet. Relat. Models, 13(6):1175–1191, 2020

    2020

  39. [47]

    Santambrogio and S

    F. Santambrogio and S. Schulz. Segregated solutions of a degenerate cross-diffusion system with drifts. arXiv preprint arXiv:2606.16367, 2026

    2026

  40. [48]

    Skrzeczkowski

    J. Skrzeczkowski. Global solutions to cross-diffusion systems with independent advections in one dimen- sion.arXiv preprint arXiv:2603.20153, 2026

    2026 arXiv

  41. [49]

    J. L. Vázquez.The porous medium equation. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. NON-UNIQUENESS OF WEAK SOLUTIONS TO CROSS-DIFFUSION SYSTEMS 33 José A. Carrillo:Mathematical Institute, University of Oxfo...

    2007

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.