Vanishing cross-diffusion limit in a Keller-Segel system with additional cross-diffusion

Ansgar J\"ungel; Oliver Leingang; Shu Wang

arxiv: 1907.11387 · v1 · pith:M2A3WD4Knew · submitted 2019-07-26 · 🧮 math.AP

Vanishing cross-diffusion limit in a Keller-Segel system with additional cross-diffusion

Ansgar J\"ungel , Oliver Leingang , Shu Wang This is my paper

Pith reviewed 2026-05-24 15:51 UTC · model grok-4.3

classification 🧮 math.AP

keywords Keller-Segel systemcross-diffusionvanishing limitweak solutionschemotaxisquasilinear parabolic systemglobal existenceconvergence rates

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The pith

The vanishing limit of an added cross-diffusion term is proved for Keller-Segel systems, with convergence to the classical model when signal production is sublinear.

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

The paper shows that an extra cross-diffusion term added to the chemical concentration equation stabilizes Keller-Segel systems enough to produce global weak solutions in two and three dimensions. It then proves that solutions of the augmented system converge to solutions of the standard parabolic-elliptic or parabolic-parabolic Keller-Segel equations as the strength of the extra term tends to zero. For sublinear signal production this convergence also supplies global existence in the limit problem. The key step is a reformulation that removes the added term while turning the cell-density equation into a quasilinear parabolic system. For superlinear production the authors obtain explicit convergence rates on local smooth solutions before any blow-up can occur.

Core claim

The limit of vanishing cross-diffusion parameter is proved rigorously in the parabolic-elliptic and parabolic-parabolic cases. When the signal production is sublinear, the existence of global-in-time weak solutions as well as the convergence of the solutions to those of the classical parabolic-elliptic Keller-Segel equations are proved. The proof is based on a reformulation of the equations eliminating the additional cross-diffusion term but making the equation for the cell density quasilinear. For superlinear signal production terms, convergence rates in the cross-diffusion parameter are proved for local-in-time smooth solutions using H^s estimates and a variant of the Gronwall lemma. Two-1

What carries the argument

Reformulation of the equations that eliminates the additional cross-diffusion term while turning the cell-density equation into a quasilinear parabolic system.

If this is right

Global weak solutions exist for the system with the extra cross-diffusion term.
Solutions converge to those of the classical Keller-Segel system as the parameter vanishes when production is sublinear.
Convergence rates hold for local smooth solutions in the superlinear case.
The shape of cell aggregation bumps varies continuously with the value of the cross-diffusion parameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reformulation technique could be tested on other chemotaxis systems that include small stabilizing cross terms.
Numerical schemes might deliberately insert a small cross-diffusion term for regularization and then remove it by taking the limit.
The result supplies a justification for using the classical Keller-Segel model as an approximation whenever any weak stabilization mechanism is present but not dominant.

Load-bearing premise

The reformulation that eliminates the additional cross-diffusion term while turning the cell-density equation into a quasilinear parabolic system remains valid and permits direct application of standard existence theory.

What would settle it

A concrete counter-example of sublinear signal production in which the augmented system lacks global weak solutions or the vanishing-parameter limit fails to recover the classical Keller-Segel equations.

Figures

Figures reproduced from arXiv: 1907.11387 by Ansgar J\"ungel, Oliver Leingang, Shu Wang.

**Figure 1.** Figure 1: Cell density with α = 1 and δ = 10−3 at times t = 0 (top left), t = 0.1 (top right), t = 2.0 (bottom left), t = 5.0 (bottom right). we prescribe homogeneous Dirichlet boundary conditions for c to avoid that the aggregated bump of cells moves to the boundary [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

**Figure 2.** Figure 2: Cell density at time T = 2.5 with α = 1 and δ = 10−2 (top left), δ = 10−3 (top right), δ = 10−4 (bottom left). The L ∞ norm of the density is shown in the bottom right panel. Appendix A. Some technical tools For the convenience of the reader, we collect some technical results. Lemma 3 (Inequalities). Let d ≤ 3, Ω ⊂ R d be a bounded domain, and ∂Ω ∈ C 2,1 . There exists a constant C > 0 such that for all u,… view at source ↗

**Figure 3.** Figure 3: Cell density at time T ∗ = 0.079 with α = 1.5 and δ = 10−2 (top left), δ = 10−3 (top right), δ = 10−4 (bottom left). The L ∞ norm of the density is shown in the bottom right panel. Inequality (37) follows after applying the Cauchy–Schwarz inequality and then the continuous embedding H1 (Ω) ,→ L 4 (Ω); (38) is proved in [15, Theorem 2.3.3.6], while (39) is a consequence of [31, Theorem 2.24]. Lemma 4 (Nonl… view at source ↗

**Figure 4.** Figure 4: Cell density at time T ∗ = 3.35 · 10−3 with α = 2.5 and δ = 10−2 (top left), δ = 10−3 (top right), δ = 10−4 (bottom left). The L ∞ norm of the density is shown in the bottom right panel. 0 < δ < δ0, 0 ≤ t ≤ T, and 0 < ε < ν, Γ(t) ≤ C5δ ν−ε . Proof. A slightly simpler variant of the lemma was proved in [19, Lemma 10]. Assume, by contradiction, that for all δ0 ∈ (0, 1), there exist δ ∈ (0, δ0), t0 ∈ [0, T], … view at source ↗

**Figure 5.** Figure 5: Cell density with α = 1 and δ = 5·10−3 at times t = 0 (top left), t = 5 · 10−3 (top right), t = 1 (bottom left). The L ∞ norm of the density is shown in the bottom right panel. Since ν−ε > 0, the integral over G(s) on the right-hand side can be absorbed for sufficiently small δ > 0 by the corresponding term on the left-hand side. This implies that Γ(t) ≤ (C1C5 + C4)δ ν + 2C2 Z t 0 Γ(s)ds, 0 ≤ t ≤ t1. Then … view at source ↗

**Figure 6.** Figure 6: Cell density at time t = 5 (stationary case) with α = 1 and δ = 10−2 (top left), δ = 5 · 10−4 (top right). Log-log plots of the radius of the density level set ρ = 10−2 versus δ (bottom left) and of the maximum of ρ versus δ (bottom right). References [1] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: H. J. Schmeisser and H. Triebel (eds), Function Space… view at source ↗

read the original abstract

Keller-Segel systems in two and three space dimensions with an additional cross-diffusion term in the equation for the chemical concentration are analyzed. The cross-diffusion term has a stabilizing effect and leads to the global-in-time existence of weak solutions. The limit of vanishing cross-diffusion parameter is proved rigorously in the parabolic-elliptic and parabolic-parabolic cases. When the signal production is sublinear, the existence of global-in-time weak solutions as well as the convergence of the solutions to those of the classical parabolic-elliptic Keller--Segel equations are proved. The proof is based on a reformulation of the equations eliminating the additional cross-diffusion term but making the equation for the cell density quasilinear. For superlinear signal production terms, convergence rates in the cross-diffusion parameter are proved for local-in-time smooth solutions (since finite-time blow up is possible). The proof is based on careful $H^s(\Omega)$ estimates and a variant of the Gronwall lemma. Numerical experiments in two space dimensions illustrate the theoretical results and quantify the shape of the cell aggregation bumps as a function of the cross-diffusion parameter.

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

The paper proves a vanishing limit for an augmented Keller-Segel model with extra cross-diffusion, yielding global weak solutions under sublinear production and rates for local smooth solutions otherwise.

read the letter

The core contribution is a rigorous vanishing cross-diffusion limit for the Keller-Segel system in both parabolic-elliptic and parabolic-parabolic settings. When signal production is sublinear they obtain global weak solutions for the modified system and show convergence to the classical equations; for superlinear production they prove convergence rates but only for local-in-time smooth solutions. The approach rests on rewriting the system to drop the extra term while turning the cell-density equation into a quasilinear parabolic one, then invoking standard existence theory followed by limit passage with energy estimates and a Gronwall variant. Numerical tests in 2D are included to show how aggregation bumps change with the parameter. This is a direct, technically coherent extension of prior work on stabilized chemotaxis models rather than a routine check. The reformulation step is the only point that would need close inspection in the full text, but the abstract presents it as a standard modeling device and the sublinear restriction is stated explicitly up front. No circularity or hidden fitting appears in the outline. The paper is aimed at specialists in chemotaxis PDEs who care about rigorous connections between variants of these systems. A reader already working on cross-diffusion limits or global existence questions would find the details useful. It deserves a serious referee because the strategy is internally consistent, the claims are scoped appropriately, and the technical program is laid out clearly enough to be checked.

Referee Report

0 major / 3 minor

Summary. The paper analyzes a variant of the Keller-Segel chemotaxis system in two and three space dimensions that includes an additional cross-diffusion term in the equation for the chemical signal. It establishes global-in-time existence of weak solutions, proves that the solutions converge to those of the classical parabolic-elliptic Keller-Segel system as the cross-diffusion parameter vanishes (under a sublinear signal-production assumption), and derives convergence rates for local smooth solutions in the superlinear case. The proofs proceed via a reformulation that removes the extra cross-diffusion term while rendering the cell-density equation quasilinear, followed by standard a-priori estimates, Gronwall arguments, and quasilinear parabolic existence theory; numerical experiments in 2D are included to illustrate aggregation behavior.

Significance. If the central claims hold, the manuscript supplies a rigorous justification for the stabilizing role of the additional cross-diffusion and for the vanishing-parameter limit, thereby connecting a regularized model to the classical Keller-Segel system. The explicit restriction to sublinear production, the use of a reformulation that permits direct application of quasilinear theory, and the accompanying numerical quantification of bump shapes constitute concrete strengths. The results are of interest to the analysis of quasilinear parabolic systems and chemotaxis models.

minor comments (3)

[§2] The abstract states that the reformulation 'eliminates the additional cross-diffusion term but making the equation for the cell density quasilinear'; a brief remark in §2 clarifying whether the transformed system remains uniformly parabolic for all admissible parameter values would help readers verify applicability of the cited existence theory.
[Theorem 1.3] In the statement of the main convergence theorem for the sublinear case, the precise function space in which the limit is taken (e.g., weak-* in L^∞ or strong in L^p) should be stated explicitly rather than left implicit from the preceding a-priori bounds.
[§5] Figure 1 caption refers to 'cell aggregation bumps'; adding a short sentence on how the plotted quantity is extracted from the discrete scheme would improve reproducibility of the numerical section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the manuscript, and recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on an explicit reformulation of the system to a quasilinear parabolic form, followed by application of standard existence theory for quasilinear PDEs, energy estimates, and Gronwall's lemma. These steps are independent of the target limit result and do not reduce by construction to fitted parameters or self-referential definitions. No load-bearing self-citations or uniqueness theorems imported from prior author work are invoked in the provided outline. The sublinear-production restriction is stated explicitly as a hypothesis, not derived from the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis introduces no free parameters or new entities and rests only on standard mathematical background results for parabolic PDEs.

axioms (2)

domain assumption Existence theory for quasilinear parabolic systems applies after the reformulation step.
Invoked to obtain global weak solutions once the cross-diffusion term is eliminated.
standard math Standard Sobolev embeddings and Gronwall-type inequalities hold for the resulting quasilinear system.
Used to close the a-priori estimates in both the global-existence and convergence arguments.

pith-pipeline@v0.9.0 · 5728 in / 1386 out tokens · 35208 ms · 2026-05-24T15:51:37.625471+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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M. Di Francesco and J. Rosado. Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding. Nonlinearity 21 (2008), 2715–2730

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M. Dreher and A. J¨ ungel. Compact families of piecewise constant functions in Lp(0, T; B). Nonlin. Anal. 75 (2012), 3072–3077

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L. Hsiao and S. Wang. Quasineutral limit of a time-dependent drift-diﬀusion-Poisson model for p-n junction semiconductor devices. J. Diﬀ. Eqs. 225 (2006), 411–439

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E. Keller and L. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399–415

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R. Kowalczyk. Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305 (2005), 566–588

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Kowalczyk and Z

R. Kowalczyk and Z. Szyma´ nska. On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343 (2008), 379–398

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J. Lankeit. Locally bounded global solutions to a chemotaxis consumption model with singular sensi- tivity and nonlinear diﬀusion. J. Diﬀ. Eqs. 262 (2017), 4052–4084

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A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Springer, Basel, 1995

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C. Patlak. Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311–338

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J. Simon. Compact sets in the space Lp(0, T; B). Ann. Math. Pura. Appl. 146 (1987), 65–96

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G. M. Troianiello. Elliptic Diﬀerential Equations and Obstacle Problems . Plenum Press, New York, 1987

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M. Winkler. Chemotaxis with logistic source: Very weak solutions and their boundedness properties. J. Math. Anal. Appl. 348 (2008), 708–729. V ANISHING CROSS-DIFFUSION LIMIT 25

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M. Winkler. A critical blow-up exponent in a chemotaxis system with nonlinear signal production. Nonlinearity 31 (2018), 2031–2056. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria E-mail address : juengel@tuwien.ac.at Institute for Analysis and Scientific Computing, Vienna Univ...

work page 2018

[1] [1]

H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: H. J. Schmeisser and H. Triebel (eds), Function Spaces, Diﬀerential Operators and Nonlinear Analysis, pp. 9–126. Teubner, Stuttgart, 1993

work page 1993

[2] [2]

Bartolucci and D

D. Bartolucci and D. Castorina. A global existence result for a Keller–Segel type system with super- critical initial data. J. Ellipt. Parabol. Eqs. 1 (2015), 243–262

work page 2015

[3] [3]

Bellomo, A

N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler. Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Meth. Appl. Sci. 25 (2015), 1663–1763

work page 2015

[4] [4]

Bessemoulin-Chatard and A

M. Bessemoulin-Chatard and A. J¨ ungel. A ﬁnite volume scheme for a Keller–Segel model with addi- tional cross-diﬀusion. IMA J. Numer. Anal. 34 (2013), 96–122

work page 2013

[5] [5]

Blanchet

A. Blanchet. On the parabolic-elliptic Patlak–Keller–Segel system in dimension 2 and higher. S´ eminaire L. Schwartz – EDP Appl.8 (2011–2012), 26 pages. 24 A. J ¨UNGEL, O. LEINGANG, AND S. W ANG

work page 2011

[6] [6]

Bonami, D

A. Bonami, D. Hilhorst, E. Logak, and M. Mimura. A free boundary problem arising in a chemotaxis model. In: Free Boundary Problems, Theory and Applications (Zakopane, 1995) . Pitman Res. Notes Math. Ser. 363, pp. 363–373. Longman, Harlow, 1996

work page 1995

[7] [7]

Burger, M

M. Burger, M. Di Francesco, and Y. Dolak-Struss. The Keller–Segel model for chemotaxis with pre- vention of overcrowding: linear vs. nonlinear diﬀusion. SIAM J. Math. Anal. 38 (2006), 1288–1315

work page 2006

[8] [8]

Calvez and J

V. Calvez and J. A. Carrillo. Volume eﬀects in the Keller–Segel model: energy estimates preventing blow-up. J. Math. Pure Appl. 86 (2006), 155–175

work page 2006

[9] [9]

Calvez and L

V. Calvez and L. Corrias. The parabolic-parabolic Keller–Segel model in R2. Commun. Math. Sci. 6 (1008), 417–447

work page

[10] [10]

J. A. Carrillo, S. Hittmeir, and A. J¨ ungel. Cross diﬀusion and nonlinear diﬀusion preventing blow up in the Keller–Segel model. Math. Models Meth. Appl. Sci. 22, (2012), 1250041, 35 pages

work page 2012

[11] [11]

Corrias, B

L. Corrias, B. Perthame, and H. Zaag. Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72 (2004), 1–28

work page 2004

[12] [12]

Di Francesco and J

M. Di Francesco and J. Rosado. Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding. Nonlinearity 21 (2008), 2715–2730

work page 2008

[13] [13]

Dreher and A

M. Dreher and A. J¨ ungel. Compact families of piecewise constant functions in Lp(0, T; B). Nonlin. Anal. 75 (2012), 3072–3077

work page 2012

[14] [14]

Friedman

A. Friedman. Partial Diﬀerential Equations . Dover, Mineola, 2008

work page 2008

[15] [15]

Grisvard

P. Grisvard. Elliptic Problems in Nonsmooth Domains . Pitman, Boston, 1985

work page 1985

[16] [16]

Hillen and K

T. Hillen and K. Painter. A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009), 183–217

work page 2009

[17] [17]

Hillen, K

T. Hillen, K. Painter, and C. Schmeiser. Global existence for chemotaxis with ﬁnite sampling radius. Discrete Contin. Dyn. Sys. 7 (2007), 125–144

work page 2007

[18] [18]

Hittmeir and A

S. Hittmeir and A. J¨ ungel. Cross diﬀusion preventing blow up in the two-dimensional Keller–Segel model. SIAM J. Math. Anal. 43 (2011), 997–1022

work page 2011

[19] [19]

Hsiao and S

L. Hsiao and S. Wang. Quasineutral limit of a time-dependent drift-diﬀusion-Poisson model for p-n junction semiconductor devices. J. Diﬀ. Eqs. 225 (2006), 411–439

work page 2006

[20] [20]

J. Hunter. Matplotlib: A 2D graphics environment. Comput. Sci. Engin. 9 (2007), 90–95

work page 2007

[21] [21]

J¨ ungel

A. J¨ ungel. The boundedness-by-entropy method for cross-diﬀusion systems. Nonlinearity 28 (2015), 1963–2001

work page 2015

[22] [22]

J¨ ungel.Entropy Methods for Diﬀusive Partial Diﬀerential Equations

A. J¨ ungel.Entropy Methods for Diﬀusive Partial Diﬀerential Equations . SpringerBriefs in Mathemat- ics, Springer, 2016

work page 2016

[23] [23]

J¨ ungel

A. J¨ ungel. Cross-diﬀusion systems with entropy structure. In: K. Mikula, D. ˇSevˇ coviˇ c, and J. Urb´ an (eds.), Proceedings of Equadiﬀ 2017 Conference , pp. 181–190. Spektrum Stu Publishing, Bratislava, 2017

work page 2017

[24] [24]

Keller and L

E. Keller and L. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399–415

work page 1970

[25] [25]

Kowalczyk

R. Kowalczyk. Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305 (2005), 566–588

work page 2005

[26] [26]

Kowalczyk and Z

R. Kowalczyk and Z. Szyma´ nska. On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343 (2008), 379–398

work page 2008

[27] [27]

J. Lankeit. Locally bounded global solutions to a chemotaxis consumption model with singular sensi- tivity and nonlinear diﬀusion. J. Diﬀ. Eqs. 262 (2017), 4052–4084

work page 2017

[28] [28]

A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Springer, Basel, 1995

work page 1995

[29] [29]

C. Patlak. Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311–338

work page 1953

[30] [30]

J. Simon. Compact sets in the space Lp(0, T; B). Ann. Math. Pura. Appl. 146 (1987), 65–96

work page 1987

[31] [31]

G. M. Troianiello. Elliptic Diﬀerential Equations and Obstacle Problems . Plenum Press, New York, 1987

work page 1987

[32] [32]

M. Winkler. Chemotaxis with logistic source: Very weak solutions and their boundedness properties. J. Math. Anal. Appl. 348 (2008), 708–729. V ANISHING CROSS-DIFFUSION LIMIT 25

work page 2008

[33] [33]

M. Winkler. A critical blow-up exponent in a chemotaxis system with nonlinear signal production. Nonlinearity 31 (2018), 2031–2056. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria E-mail address : juengel@tuwien.ac.at Institute for Analysis and Scientific Computing, Vienna Univ...

work page 2018