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arXiv · 1709.01469

On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials

Sergio Frigeri·Kei Fong Lam·Elisabetta Rocca·Giulio Schimperna
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Abstract

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 dead cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $\varphi_p$, $\varphi_d$ satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a form of the Darcy law and $n$ satisfies a quasistatic 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 view of 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.

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Sergio Frigeri, Kei Fong Lam, Elisabetta Rocca, Giulio Schimperna. 2017-09-05. On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials. https://arxiv.org/abs/1709.01469

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