Chemotaxis compressible Navier-Stokes equations with density-dependent viscosity modeling vascular network formation

Ansgar J\"ungel; Flora Philipp

arxiv: 2604.24725 · v1 · submitted 2026-04-27 · 🧮 math.AP

Chemotaxis compressible Navier-Stokes equations with density-dependent viscosity modeling vascular network formation

Ansgar J\"ungel , Flora Philipp This is my paper

Pith reviewed 2026-05-08 02:00 UTC · model grok-4.3

classification 🧮 math.AP

keywords chemotaxiscompressible Navier-Stokesweak solutionsvascular network formationdensity-dependent viscosityendothelial cellsglobal existenceKeller-Segel system

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The pith

Global weak solutions exist for the compressible Navier-Stokes equations with chemotaxis that model early vascular network formation.

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

The paper establishes the existence of global weak solutions to a coupled system consisting of compressible Navier-Stokes equations for endothelial cell density and velocity together with a reaction-diffusion equation for the chemoattractant concentration. The system is posed on the three-dimensional torus and the result applies to initial data with finite energy when the adiabatic pressure exponent satisfies gamma greater than 4/3. The proof proceeds by adding Korteweg and drag regularization terms, deriving uniform bounds from the BD entropy inequality, and passing to the limit while preserving a renormalized formulation in the velocity. A sympathetic reader would care because the result shows that the model remains well-defined over infinite time for a physically relevant range of parameters, supplying a mathematical foundation for describing self-assembly of cells into blood vessels without immediate singularity formation.

Core claim

The existence of global weak solutions to the compressible Navier-Stokes equations for the density of endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, is established in a three-dimensional torus for energy-finite initial data. The coupling of the equations arises through the chemotaxis force, which contributes to the momentum balance equation, and the signal production due to the cells in the chemotaxis equation. The equations model the self-assembly of endothelial cells during the early stages of blood vessel formation. The existence result holds for adiabatic pressure exponents gamma greater than 4/3. The proof is,

What carries the argument

Korteweg-drag approximation together with the BD entropy inequality, which supplies uniform bounds and permits passage to the limit while constructing renormalized weak solutions in the velocity variable.

If this is right

Global-in-time weak solutions exist without finite-time blow-up for all energy-finite initial data when gamma exceeds 4/3.
The weak solutions satisfy a renormalized formulation with respect to the velocity field.
The existence range gamma greater than 4/3 matches the threshold known for the degenerate Keller-Segel system without fluid coupling.
The construction applies directly to the modeling of endothelial cell self-assembly on periodic domains.

Where Pith is reading between the lines

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

The same approximation and entropy strategy could be tested on related models that include cell proliferation or matrix degradation terms.
Numerical schemes built from the regularized system may inherit long-time stability from the proven bounds.
Lowering the exponent threshold to gamma equals or less than 4/3 would require either a different viscosity law or additional a-priori estimates beyond the BD inequality.
The weak solutions provide a starting point for studying the emergence of network-like structures through passage to a free-boundary limit.

Load-bearing premise

Initial data must possess finite energy and the density-dependent viscosity must be chosen so that the BD entropy inequality survives the Korteweg-drag regularization.

What would settle it

A concrete set of finite-energy initial data on the three-dimensional torus for which the regularized solutions fail to converge to a limit satisfying the original equations when gamma equals 4/3 would disprove the existence claim.

read the original abstract

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

They prove global weak solutions for the compressible NS-chemotaxis system modeling vascular formation on the torus with gamma > 4/3 via BD entropy and Korteweg-drag approximation.

read the letter

The main result is the existence of global weak solutions to this compressible Navier-Stokes system coupled with chemotaxis, modeling the early stages of blood vessel formation by endothelial cells. They work on the 3D torus with finite energy data and gamma > 4/3. The novelty lies in handling the density-dependent viscosity together with the chemotactic forcing in the fluid equations. Prior work on Keller-Segel is degenerate but without the full fluid dynamics, and standard compressible NS lacks the biology coupling. The proof adapts known entropy techniques to this setting. They do well by using the Korteweg-drag approximation to get regularized equations where the BD entropy inequality holds uniformly. This gives control on the density and the velocity gradients despite the degeneracy. The renormalized velocity formulation then helps pass to the limit in the nonlinear coupling terms with the chemoattractant equation. The argument seems free of circularity and the limit passages look manageable based on the structure described. No internal contradictions jump out. A soft spot might be the verification that the limiting objects satisfy the original equations, especially with the density-dependent viscosity affecting the stress tensor. But the paper claims the renormalized solutions work, so this is probably checked. This is targeted at mathematicians in applied PDEs and mathematical biology. Someone studying existence for fluid models with biological interactions would get value from the technical details. It deserves peer review as a careful extension of existing methods to a relevant model. I would recommend sending it to a journal in the field for full evaluation.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes the existence of global weak solutions to the compressible Navier-Stokes equations governing endothelial cell density and velocity, coupled to a reaction-diffusion equation for the chemoattractant concentration, on the three-dimensional torus. The result applies to finite-energy initial data when the adiabatic exponent satisfies γ > 4/3. The proof relies on a Korteweg-drag regularization, uniform bounds from the BD entropy inequality, and passage to the limit in a renormalized velocity formulation.

Significance. If the result holds, it supplies the first global existence theorem for a density-dependent-viscosity fluid model of early vascular network formation that incorporates both chemotactic forcing and signal production. The threshold γ > 4/3 matches the known threshold for the degenerate Keller-Segel system, and the BD entropy supplies the key integrability that controls the degeneracy without additional structural assumptions. This is a concrete advance at the interface of compressible fluid dynamics and chemotaxis.

major comments (2)

The outline indicates that the BD entropy inequality survives the Korteweg-drag approximation and yields uniform bounds on density and velocity gradients. However, the precise form of the drag term and its interaction with the density-dependent viscosity coefficient must be verified explicitly so that the entropy production terms remain non-negative and independent of the approximation parameter; otherwise the passage to the limit in the momentum equation cannot close.
In the limit passage for the chemotaxis force term (density times gradient of chemoattractant) and the signal-production term, the renormalized velocity formulation is invoked to recover compactness. The argument requires a concrete identification of the limit in the product of the (only L^γ) density with the velocity and with the chemoattractant gradient; without an explicit strong-convergence or compensated-compactness step for these products, the weak solution may fail to satisfy the original equations.

minor comments (2)

Notation for the renormalized velocity should be introduced with an explicit definition (e.g., the precise test-function class or the form of the renormalized continuity equation) already in the statement of the main theorem.
The manuscript should include a short remark comparing the obtained regularity (especially the integrability of the velocity gradient) with the corresponding regularity in the non-fluid degenerate Keller-Segel literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major points below and will incorporate additional explicit verifications and details into the revised version.

read point-by-point responses

Referee: The outline indicates that the BD entropy inequality survives the Korteweg-drag approximation and yields uniform bounds on density and velocity gradients. However, the precise form of the drag term and its interaction with the density-dependent viscosity coefficient must be verified explicitly so that the entropy production terms remain non-negative and independent of the approximation parameter; otherwise the passage to the limit in the momentum equation cannot close.

Authors: We appreciate the referee drawing attention to the need for explicit verification. The drag term is introduced as a linear friction term proportional to the velocity, chosen precisely so that its contribution to the BD entropy production is non-negative and cancels appropriately with the viscous dissipation terms involving the density-dependent viscosity. In the current manuscript this is indicated in the derivation of the approximate entropy inequality, but we agree that the cross terms and their independence of the regularization parameter should be written out in full. In the revised manuscript we will add a dedicated lemma containing the complete expansion of the entropy production, confirming non-negativity and uniformity. revision: yes
Referee: In the limit passage for the chemotaxis force term (density times gradient of chemoattractant) and the signal-production term, the renormalized velocity formulation is invoked to recover compactness. The argument requires a concrete identification of the limit in the product of the (only L^γ) density with the velocity and with the chemoattractant gradient; without an explicit strong-convergence or compensated-compactness step for these products, the weak solution may fail to satisfy the original equations.

Authors: We agree that the identification of the nonlinear terms deserves a more detailed exposition. The renormalized velocity formulation, together with the strong compactness of the chemoattractant concentration (obtained from the parabolic regularity of the reaction-diffusion equation and the uniform L^∞ bound on c), permits passage to the limit in the products ρ ∇c and ρ f(c) via a compensated-compactness argument that exploits the effective viscous flux identity derived from the BD entropy. In the revised manuscript we will insert an explicit subsection that carries out the limit identification step by step, specifying the test functions used in the renormalized formulation and the convergence modes employed for each factor. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard techniques

full rationale

The paper constructs global weak solutions through a Korteweg-drag approximation, uniform BD entropy bounds derived from the system structure, and limit passage using renormalized velocity formulations and compactness from the chemoattractant equation. These steps rely on independent a priori estimates and standard functional-analytic tools (e.g., Aubin-Lions, renormalized solutions) that do not presuppose the target existence result or reduce to fitted parameters. No self-definitional relations, load-bearing self-citations, or ansatz smuggling appear in the provided derivation chain; the γ > 4/3 threshold emerges directly from the entropy integrability without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard PDE theory (weak convergence, entropy inequalities) and domain assumptions without free parameters or new postulated entities.

axioms (2)

standard math Standard functional-analytic tools (Sobolev embeddings, weak compactness) apply to the approximated system on the torus.
Invoked to pass to the limit after regularization.
domain assumption The BD entropy inequality remains valid for the regularized equations with the chosen density-dependent viscosity.
Central to obtaining uniform bounds independent of the approximation parameter.

pith-pipeline@v0.9.0 · 5440 in / 1344 out tokens · 58938 ms · 2026-05-08T02:00:26.271232+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 2 canonical work pages

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Y. Tao and M. Winkler. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity.J. Differ. Eqs.252 (2012), 692–715

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Vasseur and C

A. Vasseur and C. Yu. Global weak solutions to the compressible quantum Navier–Stokes equations with damping.SIAM J. Math. Anal.48 (2016), 1489–1511

2016
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Vasseur and C

A. Vasseur and C. Yu. Existence of global weak solutions for 3D degenerate compressible Navier– Stokes equations.Invent. Math.206 (2016), 935–974

2016
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M. Winkler. Does a ‘volume-filling effect’ always prevent chemotactic collapse?Math. Meth. Appl. Sci.33 (2010), 12–24

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Zatorska

E. Zatorska. On the flow of chemically reacting gaseous mixture.J. Differ. Eqs.253 (2012), 3471–3500. Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address:juengel@tuwien.ac.at Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address:fl...

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[1] [1]

A¨ ıssa and R

N. A¨ ıssa and R. Alexandre. Global existence of weak solutions to an angiogenesis model.J. Evol. Eqs. 16 (2016), 877–894

2016

[2] [2]

Ambrosi, A

D. Ambrosi, A. Gamba, and G. Serini. Cell directional and chemotaxis in vascular morphogenesis. Bull. Math. Biol.66 (2004), 1851–1873

2004

[3] [3]

Antonelli, D

P. Antonelli, D. Bresch, and S. Spirito. Global weak solutions to the Navier–Stokes–Korteweg equa- tions in one dimension. Submitted for publication, 2025. arXiv:2502.17147

work page arXiv 2025

[4] [4]

Antonelli and S

P. Antonelli and S. Spirito. On the compactness of finite energy weak solutions to the quantum Navier–Stokes equations.J. Hyperbolic Differ. Eqs.15 (2018), 133–147

2018

[5] [5]

Bresch and B

D. Bresch and B. Desjardins. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model.Commun. Math. Phys.238 (2003), 211– 223

2003

[6] [6]

Bresch and B

D. Bresch and B. Desjardins. On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier–Stokes models.J. Math. Pures Appl.86 (2006), 362–368

2006

[7] [7]

Bresch, B

D. Bresch, B. Desjardins, and C.-K. Lin. On some compressible fluid models: Korteweg, lubrication, and shallow water systems.Commun. Partial Differ. Eqs.28 (2003), 843–868

2003

[8] [8]

Bresch, A

D. Bresch, A. Vasseur, and C. Yu. Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities.J. Eur. Math. Soc. 24 (2022), 1791–1837

2022

[9] [9]

Deshmukh, K

K. Deshmukh, K. Mitra, and A. Bit. Influence of non-Newtonian viscosity on flow structures and wall deformation in compliant serpentine microchannels: a numerical study.Micromachines14 (2023), no. 1661, 16 pages

2023

[10] [10]

Feireisl.Dynamics of Viscous Compressible Fluids

E. Feireisl.Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford, 2004

2004

[11] [11]

Feireisl, A

E. Feireisl, A. Novotn´ y, and H. Petzeltov´ a. On the existence of globally defined weak solutions to the Navier–Stokes equations.J. Math. Fluid Mech.3 (2001), 358–392

2001

[12] [12]

Huo and A

X. Huo and A. J¨ ungel. Global existence and weak–strong uniqueness for chemotaxis compressible Navier–Stokes equations modeling vascular network formation.J. Math. Fluid Mech.26 (2024), no. 11, 19 pages

2024

[13] [13]

Jiang and P

S. Jiang and P. Zhang. Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids.J. Math. Pures Appl.82 (2003), 949–973

2003

[14] [14]

J¨ ungel

A. J¨ ungel. Global weak solutions to compressible Navier–Stokes equations for quantum fluids.SIAM J. Math. Anal.42 (2010), 1025–1045

2010

[15] [15]

Lacroix-Violet and A

I. Lacroix-Violet and A. Vasseur. Global weak solutions to the compressible quantum Navier–Stokes equation and its semi-classical limit.J. Math. Pures Appl.114 (2018), 191–210

2018

[16] [16]

Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

J. Li and Z. Xin. Global existence of weak solutions to the barotropic compressible Navier–Stokes flows with degenerate viscosities. arXiv:1504.06826, 2015

work page Pith review arXiv 2015

[17] [17]

Lions.Mathematical Topics in Fluid Mechanics, Vol

P.-L. Lions.Mathematical Topics in Fluid Mechanics, Vol. 2: Compressible Models. Clarendon Press, Oxford, 1998

1998

[18] [18]

Mellet and A

A. Mellet and A. Vasseur. On the barotropic compressible Navier–Stokes equations.Commun. Partial Differ. Eqs.32 (2007), 431–452

2007

[19] [19]

Novotn´ y and I

A. Novotn´ y and I. Straˇ skraba.Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford, 2004. CHEMOTAXIS COMPRESSIBLE NA VIER–STOKES EQUATIONS 27

2004

[20] [20]

Sugiyama

Y. Sugiyama. Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis.Differ. Integral Eqs.20 (2007), 133–180

2007

[21] [21]

Tao and M

Y. Tao and M. Winkler. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity.J. Differ. Eqs.252 (2012), 692–715

2012

[22] [22]

Vasseur and C

A. Vasseur and C. Yu. Global weak solutions to the compressible quantum Navier–Stokes equations with damping.SIAM J. Math. Anal.48 (2016), 1489–1511

2016

[23] [23]

Vasseur and C

A. Vasseur and C. Yu. Existence of global weak solutions for 3D degenerate compressible Navier– Stokes equations.Invent. Math.206 (2016), 935–974

2016

[24] [24]

M. Winkler. Does a ‘volume-filling effect’ always prevent chemotactic collapse?Math. Meth. Appl. Sci.33 (2010), 12–24

2010

[25] [25]

Zatorska

E. Zatorska. On the flow of chemically reacting gaseous mixture.J. Differ. Eqs.253 (2012), 3471–3500. Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address:juengel@tuwien.ac.at Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address:fl...

2012