The relaxation of the Cahn-Hilliard equation for the modelling of tumors and its numerical simulation (Alexandre Poulain, Simula Research Laboratory, Oslo)

Séminaire « Analyse numérique et équations aux dérivées partielles »
Salle de réunion, bâtiment M2

The Cahn-Hilliard equation, arising from physics, describes the phase separa-
tion occurring in a material during a sudden cooling process and is the subject of
many pieces of research [2]. An interesting application of this equation is its ca-
pacity to model cell populations undergoing attraction and repulsion effects [4].
For this application, we consider a variant of the Cahn-Hilliard equation with
a single-well potential and a degenerate mobility [1]. This particular form in-
troduces numerous difficulties especially for numerical simulations. We propose
a relaxation of the equation to tackle these issues and analyze the resulting
system [3]. Interestingly, this relaxed version of the degenerate Cahn-Hilliard
equation bears some similarity with a nonlinear Keller-Segel model. We also
describe a simple finite element scheme that preserves the critical physical (or
biological) properties using an upwind approach. References [1] A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli, and M. Verani, A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), pp. 7598–7626. [2] A. Miranville, The Cahn-Hilliard equation, vol. 95 of CBMS-NSF Re- gional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. Recent advances and applications. [3] B. Perthame and A. Poulain, Relaxation of the Cahn-Hilliard equa- tion with singular single-well potential and degenerate mobility, European J. Appl. Math., 32 (2021), pp. 89–112. [4] S. M. Wise, J. S. Lowengrub, H. B. Frieboes, and V. Cristini, Three-dimensional multispecies nonlinear tumor growth—I: Model and nu- merical method, J. Theoret. Biol., 253 (2008), pp. 524–543. 

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