Abstract
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.
Highlights
In recent years there has been an increased focus on the mathematical modelling and analysis of tumour growth
In this work we analyse a diffuse interface model proposed in [20], which models a mixture of tumour cells and healthy cells in the presence of an unspecified chemical species acting as a nutrient
We refer the reader to [20, §2] for the derivation from thermodynamic principles, and to [20, §2.5] for a discussion regarding the choices for the source terms Γφ, Γv and S
Summary
In recent years there has been an increased focus on the mathematical modelling and analysis of tumour growth. The well-posedness of model (1.5) with the choice (1.7) has been studied by the authors in [17] and [18] with the boundary conditions (1.2) (neglecting (1.2b)) in the former and for non-zero Dirichlet boundary conditions in the latter It has been noted in [17] that the wellposedness result with the boundary conditions (1.2) requires Ψ to have at most quadratic growth, which is attributed to the presence of the source term Γφμ = h(φ)μ(λpσ−λa) when deriving useful a priori estimates. We are able to prove existence of weak solutions for Γφ of the form (2.1), which generalises the choices (1.6) and (1.7), but in exchange Γv has to be considered as a prescribed function This is attributed to the presence of the source term Γv φμ.
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