Analysis of a diffuse interface model of multispecies tumor growth. (9th March 2017)

• Record Type: Journal Article
• Authors: Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Schonbek, Maria E
• Is Part Of: Nonlinearity. Volume 30:Number 4(2017:Apr.)
• Page Start: 1639
• Page End: 1658
• Publication Date: 2017-03-09
• Subjects: tumor growth -- diffuse interface model -- Cahn–Hilliard equation -- reaction-diffusion equation -- Darcy law -- weak solution -- singular limits; 35B25 -- 35D30 -- 35K35 -- 35K57 -- 35Q92 -- 74G25 -- 78A70 -- 92C17; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa6063
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We consider a diffuse interface model for tumor growth recently proposed in Chen et al (2014 Int. J. Numer. Methods Biomed. Eng . 30 726–54). In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn–Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity u satisfies u ⋅ ν > 0, where ν is the outer normal to the boundary of the domain.