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AIMS Mathematics, 2016, 1(3): 318-360. doi: 10.3934/Math.2016.3.318. Research article Special Issue

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Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth

Harald Garcke, Kei Fong Lam

Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany

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Special Issue: Nonlinear Evolution PDEs, Interfaces and Applications

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We study the existence of weak solutions to a Cahn-Hilliard-Darcy system coupled witha convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’slaw. The system of equations arises from a mixture model for tumour growth accounting for transportmechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, theexistence of global weak solutions in two and three dimensions, along with new regularity results forthe velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivativeshave lower regularity compared to systems without Darcy flow, but in the two dimensional case weemploy a new regularity result for the velocity to obtain better integrability and temporal regularityfor the time derivatives. Then, we deduce the global existence of weak solutions for two variantsof the model; one where the velocity is zero and another where the chemotaxis and active transportmechanisms are absent.

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Keywords: Cahn-Hilliard-Darcy system; phase field model; convection-reaction-diffusion equation; tumour growth; chemotaxis; weak solutions; asymptotic analysis

Citation: Harald Garcke, Kei Fong Lam. Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Mathematics, 2016, 1(3): 318-360. doi: 10.3934/Math.2016.3.318

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This article has been cited by:

1. Harald Garcke, Kei Fong Lam, Elisabetta Rocca, Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth, Applied Mathematics & Optimization, 2017, 10.1007/s00245-017-9414-4
2. Harald Garcke, Kei Fong Lam, Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete and Continuous Dynamical Systems, 2017, 37, 8, 4277, 10.3934/dcds.2017183
3. Luca Dedè, Harald Garcke, Kei Fong Lam, A Hele–Shaw–Cahn–Hilliard Model for Incompressible Two-Phase Flows with Different Densities, Journal of Mathematical Fluid Mechanics, 2017, 10.1007/s00021-017-0334-5

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Copyright Info: © 2016, Harald Garcke, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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