Nonlocal Cahn-Hilliard equation with degenerate mobility: Incompressible limit and convergence to stationary states - Archive ouverte HAL
Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2024

Nonlocal Cahn-Hilliard equation with degenerate mobility: Incompressible limit and convergence to stationary states

Résumé

The link between compressible models of tissue growth and the Hele-Shaw free boundary problem of fluid mechanics has recently attracted a lot of attention. In most of these models, only repulsive forces and advection terms are taken into account. In order to take into account long range interactions, we include for the first time a surface tension effect by adding a nonlocal term which leads to the degenerate nonlocal Cahn-Hilliard equation, and study the incompressible limit of the system. The degeneracy and the source term are the main difficulties. Our approach relies on a new L ∞ estimate obtained by De Giorgi iterations and on a uniform control of the energy despite the source term. We also prove the long-term convergence to a single constant stationary state of any weak solution using entropy methods, even when a source term is present. Our result shows that the surface tension in the nonlocal (and even local) Cahn-Hilliard equation will not prevent the tumor from completely invading the domain.
Fichier principal
Vignette du fichier
CH_incompressible_limit_nonlocal.pdf (669.61 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04094130 , version 1 (10-05-2023)

Identifiants

Citer

Charles Elbar, Benoît Perthame, Andrea Poiatti, Jakub Skrzeczkowski. Nonlocal Cahn-Hilliard equation with degenerate mobility: Incompressible limit and convergence to stationary states. Archive for Rational Mechanics and Analysis, 2024, 248 (3), pp.41. ⟨10.1007/s00205-024-01990-0⟩. ⟨hal-04094130⟩
40 Consultations
25 Téléchargements

Altmetric

Partager

More