Communications in Mathematical Sciences
Volume 16 (2018)
On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials
Pages: 821 – 856
We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $\varphi_p , \varphi_d$ (proliferating and necrotic cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $\varphi_p , \varphi_d$ satisfy a vectorial Cahn–Hilliard-type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a variant of Darcy’s law and $n$ satisfies a quasi-static diffusion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector ($\varphi_p , \varphi_d$) is constrained to remain in the range of physically admissible values. On the other hand, in the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of $\varphi_p$ and $\varphi_d$. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.
tumor growth, nonlinear evolutionary system, Cahn–Hilliard–Darcy system, existence of weak solutions, logarithmic potentials
2010 Mathematics Subject Classification
35D30, 35K57, 35Q35, 35Q92, 76S05, 92B05, 92C17
This research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese”. The present paper also benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations” for GS, and of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for SF, ER, and GS. SF is “titolare di un Assegno di Ricerca dell’Istituto Nazionale di Alta Matematica”.
Received 5 September 2017
Received revised 6 February 2018
Accepted 6 February 2018
Published 30 August 2018