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Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport

Published online by Cambridge University Press:  29 June 2016

HARALD GARCKE  and
KEI FONG LAM
HARALD GARCKE
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany email: Harald.Garcke@mathematik.uni-regensburg.de, Kei-Fong.Lam@mathematik.uni-regensburg.de
KEI FONG LAM
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany email: Harald.Garcke@mathematik.uni-regensburg.de, Kei-Fong.Lam@mathematik.uni-regensburg.de
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Abstract

We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

Keywords

Tumour growthphase field modelCahn–Hilliard equationreaction-diffusion equationschemotaxisweak solutionswell-posedness
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 2 , April 2017 , pp. 284 - 316
Copyright
Copyright © Cambridge University Press 2016 

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