Improved convergence rates for the Hele-Shaw limit in the presence of confining potentials

Noemi David; Alpár R. Mészáros; Filippo Santambrogio

doi:10.5802/jep.322

Improved convergence rates for the Hele-Shaw limit in the presence of confining potentials
[Meilleurs taux de convergence pour la limite de Hele-Shaw en présence de potentiels de confinement]

Noemi David ¹ ; Alpár R. Mészáros ² ; Filippo Santambrogio ³

¹ CNRS, Laboratoire de Mathématiques Raphaël Salem, Avenue de l’Université, BP.12, 76801, Saint-Étienne-du-Rouvray, France
² Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom
³ Universite Claude Bernard Lyon 1, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, Institut Camille Jordan, UMR5208, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex

Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 41-71

Abstract (VO)
Résumé

Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modeling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. In this paper, we address the question of estimating the rate of this asymptotics in the presence of external drifts. In particular, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials.

Il existe aujourd’hui une vaste littérature sur la limite incompressible pour des équations de diffusion non linéaires vers leur limite de type Hele-Shaw. Ce problème a suscité beaucoup d’intérêt en raison de ses applications à la croissance des tissus et à la modélisation des mouvements de foule, car il permet de relier des modèles compressibles à ceux, incompressibles, où les effets de congestion sont traités par des contraintes. Dans cet article, nous abordons la question de l’estimation du taux de cette convergence en présence d’advection externe. En particulier, nous fournissons des résultats améliorés dans la distance Wasserstein-2 qui sont globaux en temps grâce à ses propriétés de contractivité lorsque les potentiels sont strictement convexes.

Reçu le : 2025-01-05
Accepté le : 2025-10-16
Publié le : 2025-11-04

DOI : 10.5802/jep.322

Classification : 35B45, 35K65, 35Q92, 49Q22, 76N10, 76T99
Keywords: Porous medium equation, incompressible limit, convergence rate, Hele-Shaw free boundary problem, Wasserstein metric, gradient flow
Mots-clés : Équation des milieux poreux, limite incompressible, taux de convergence, problème à frontière libre de Hele-Shaw, métrique de Wasserstein, flot de gradient

Affiliations des auteurs :

Noemi David ¹ ; Alpár R. Mészáros ² ; Filippo Santambrogio ³

¹ CNRS, Laboratoire de Mathématiques Raphaël Salem, Avenue de l’Université, BP.12, 76801, Saint-Étienne-du-Rouvray, France
² Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom
³ Universite Claude Bernard Lyon 1, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, Institut Camille Jordan, UMR5208, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Noemi David and Alp\'ar R. M\'esz\'aros and Filippo Santambrogio},
     title = {Improved convergence rates for {the~Hele-Shaw} limit in the presence of confining~potentials},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {41--71},
     year = {2026},
     publisher = {\'Ecole polytechnique},
     volume = {13},
     doi = {10.5802/jep.322},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.322/}
}

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Noemi David; Alpár R. Mészáros; Filippo Santambrogio. Improved convergence rates for the Hele-Shaw limit in the presence of confining potentials. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 41-71. doi: 10.5802/jep.322

Bibliographie
Cité par

[AGS08] L. Ambrosio, N. Gigli & G. Savaré - Gradient flows in metric spaces and in the space of probability measures, Lectures in Math. ETH Zürich, Birkhäuser Verlag, 2008 | MR | Zbl

[AKY14] D. Alexander, I. Kim & Y. Yao - “Quasi-static evolution and congested crowd transport”, Nonlinearity 27 (2014) no. 4, p. 823-858 | DOI | Zbl | MR

[BDDR08] F. Berthelin, P. Degond, M. Delitala & M. Rascle - “A model for the formation and evolution of traffic jams”, Arch. Rational Mech. Anal. 187 (2008) no. 2, p. 185-220 | MR | Zbl | DOI

[BGG12] F. Bolley, I. Gentil & A. Guillin - “Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations”, J. Funct. Anal. 263 (2012) no. 8, p. 2430-2457 | MR | DOI | Zbl

[BGG13] F. Bolley, I. Gentil & A. Guillin - “Uniform convergence to equilibrium for granular media”, Arch. Rational Mech. Anal. 208 (2013) no. 2, p. 429-445 | Zbl | MR | DOI

[BJR07] G. Bouchitté, C. Jimenez & M. Rajesh - “A new $L^{\infty}$ estimate in optimal mass transport”, Proc. Amer. Math. Soc. 135 (2007) no. 11, p. 3525-3535 | DOI | MR | Zbl

[BPPS20] F. Bubba, B. Perthame, C. Pouchol & M. Schmidtchen - “Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues”, Arch. Rational Mech. Anal. 236 (2020), p. 735-766 | DOI | MR

[CCY19] J. A. Carrillo, K. Craig & Y. Yao - “Aggregation-diffusion equations: dynamics, asymptotics, and singular limits”, in Active particles. Vol. 2. Advances in theory, models, and applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2019, p. 65-108 | Zbl

[CG21] J. A. Carrillo & R. S. Gvalani - “Phase transitions for nonlinear nonlocal aggregation-diffusion equations”, Comm. Math. Phys. 382 (2021) no. 1, p. 485-545 | DOI | MR | Zbl

[CKY18] K. Craig, I. Kim & Y. Yao - “Congested aggregation via Newtonian interaction”, Arch. Rational Mech. Anal. 227 (2018) no. 1, p. 1-67 | DOI | MR

[CMV06] J. A. Carrillo, R. J. McCann & C. Villani - “Contractions in the 2-Wasserstein length space and thermalization of granular media”, Arch. Rational Mech. Anal. 179 (2006) no. 2, p. 217-263 | DOI | MR | Zbl

[CT20] K. Craig & I. Topaloglu - “Aggregation-diffusion to constrained interaction: minimizers & gradient flows in the slow diffusion limit”, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020) no. 2, p. 239-279 | DOI | MR | Zbl | Numdam

[DDP23] N. David, T. Dębiec & B. Perthame - “Convergence rate for the incompressible limit of nonlinear diffusion-advection equations”, Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023) no. 3, p. 511-529 | Numdam | Zbl | MR | DOI

[DH13] P. Degond & J. Hua - “Self-organized hydrodynamics with congestion and path formation in crowds”, J. Comput. Phys. 237 (2013), p. 299-319 | DOI | Zbl | MR

[DHV20] P. Degond, S. Hecht & N. Vauchelet - “Incompressible limit of a continuum model of tissue growth for two cell populations”, Netw. Heterog. Media 15 (2020) no. 1, p. 57-85 | DOI | Zbl | MR

[DMM16] S. Di Marino & A. R. Mészáros - “Uniqueness issues for evolution equations with density constraints”, Math. Models Methods Appl. Sci. 26 (2016) no. 9, p. 1761-1783 | DOI | Zbl | MR

[DPSV21] T. Dębiec, B. Perthame, M. Schmidtchen & N. Vauchelet - “Incompressible limit for a two-species model with coupling through Brinkman’s law in any dimension”, J. Math. Pures Appl. (9) 145 (2021), p. 204-239 | DOI | MR | Zbl

[DS20] T. Dębiec & M. Schmidtchen - “Incompressible limit for a two-species tumour model with coupling through Brinkman’s law in one dimension”, Acta Appl. Math. 169 (2020), p. 593-611 | DOI | MR | Zbl

[DS24] N. David & M. Schmidtchen - “On the incompressible limit for a tumour growth model incorporating convective effects”, Comm. Pure Appl. Math. 77 (2024) no. 5, p. 2577-2859 | MR | Zbl

[EG92] L. Evans & R. Gariepy - Measure theory and fine properties of functions, Studies in Advanced Math., CRC Press, Boca Raton, FL, 1992 | Zbl | MR

[HLP23] Q. He, H.-L. Li & B. Perthame - “Incompressible limits of the Patlak-Keller-Segel model and its stationary state”, Acta Appl. Math. 188 (2023), article ID 11, 53 pages | MR | Zbl

[HV17] S. Hecht & N. Vauchelet - “Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint”, Commun. Math. Sci. 15 (2017) no. 7, p. 1913-1932 | DOI | Zbl | MR

[KPW19] I. Kim, N. Požár & B. Woodhouse - “Singular limit of the porous medium equation with a drift”, Adv. Math. 349 (2019), p. 682-732 | DOI | MR | Zbl

[LX21] J.-G. Liu & X. Xu - “Existence and incompressible limit of a tissue growth model with autophagy”, SIAM J. Math. Anal. 53 (2021) no. 5, p. 5215–5242 | MR | Zbl

[MRCS10] B. Maury, A. Roudneff-Chupin & F. Santambrogio - “A macroscopic crowd motion model of gradient flow type”, Math. Models Methods Appl. Sci. 20 (2010) no. 10, p. 1787-1821 | MR | Zbl | DOI

[MRCSV11] B. Maury, A. Roudneff-Chupin, F. Santambrogio & J. Venel - “Handling congestion in crowd motion modeling”, Netw. Heterog. Media 6 (2011) no. 3, p. 485-519 | DOI | MR | Zbl

[MS16] A. R. Mészáros & F. Santambrogio - “Advection-diffusion equations with density constraints”, Anal. PDE 9 (2016) no. 3, p. 615-644 | DOI | MR | Zbl

[Ott01] F. Otto - “The geometry of dissipative evolution equations: the porous medium equation”, Comm. Partial Differential Equations 26 (2001) no. 1-2, p. 101-174 | DOI | MR | Zbl

[Pey18] R. Peyre - “Comparison between $W_{2}$ distance and ${\dot{H}}^{- 1}$ norm, and localization of Wasserstein distance”, ESAIM Control Optim. Calc. Var. 24 (2018) no. 4, p. 1489-1501 | MR | Zbl | DOI

[PQV14] B. Perthame, F. Quirós & J. L. Vázquez - “The Hele-Shaw asymptotics for mechanical models of tumor growth”, Arch. Rational Mech. Anal. 212 (2014) no. 1, p. 93-127 | DOI | MR | Zbl

[PV15] B. Perthame & N. Vauchelet - “Incompressible limit of a mechanical model of tumour growth with viscosity”, Philos. Trans. Roy. Soc. A 373 (2015) no. 2050, article ID 20140283, 16 pages | MR | Zbl

[San15] F. Santambrogio - Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their App., vol. 87, Birkhäuser/Springer, Cham, 2015 | DOI | MR | Zbl

[Vil03] C. Villani - Topics in optimal transportation, Graduate Studies in Math., vol. 58, American Mathematical Society, Providence, RI, 2003 | DOI | MR | Zbl

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