Comptes Rendus
Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 375-394.

The Cahn–Hilliard equation with degenerate mobility is used in several areas including the modeling of living tissues, following the theory of mixtures. We are interested in quantifying the pressure jump at the interface between phases in the case of incompressible flows. To do so, we depart from the spherically symmetric dynamical compressible model and include an external force. We prove existence of stationary states as limits of the parabolic problems. Then we prove the incompressible limit and characterize compactly supported stationary solutions. This allows us to compute the pressure jump in the small dispersion regime and in particular the force dependent curvature effect.

L’équation de Cahn–Hilliard avec mobilité dégénérée est utilisée dans différents domaines, en particulier la description de tissus vivants suivant la théorie des mélanges. Nous visons à quantifier le saut de pression à l’interface entre phases dans le cas de flots incompressibles. Pour cela, nous considérons des solutions à symmétrie radiale du problème compressible. Nous démontrons l’existence d’états stationnaires comme limite du problème d’évolution. Nous prouvons ensuite la limite incompressible et caratérisons les solutions à support compact. Ceci nous permet de calculer le saut de pression dans le régime de faible dispersion et en particulier d’obtenir la dépendance en la courbure suivant la force appliquée.

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DOI: 10.5802/crmeca.173
Classification: 35B40, 35B45, 35G20, 35Q92
Keywords: Degenerate Cahn–Hilliard equation, Asymptotic Analysis, Incompressible limit, Hele–Shaw equations, Surface tension, Pressure jump
Mot clés : Equation de Cahn–Hilliard dégénérée, Analyse asymptotique, Limite incompressible, Equation de Hele–Shaw, Saut de pression

Charles Elbar 1; Benoît Perthame 1; Jakub Skrzeczkowski 2

1 Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions, F-75005 Paris, France
2 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Stefana Banacha 2, 02-097 Warsaw, Poland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Charles Elbar; Benoît Perthame; Jakub Skrzeczkowski. Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 375-394. doi : 10.5802/crmeca.173. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.173/

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