WIAS Preprint No. 2150, (2015)

Analysis of a diffuse interface model of multispecies tumor growth


  • Dai, Mimi
  • Feireisl, Eduard
  • Rocca, Elisabetta
  • Schimperna, Giulio
  • Schonbek , Maria E.

2010 Mathematics Subject Classification

  • 35B25 35D30 35K35 35K57 35Q92 74G25 78A70 92C17


  • tumor growth, diffuse interface model, Cahn-Hilliard equation, reaction-diffusion equation, Darcy law, existence of weak solutions, singular limits




We consider a diffuse interface model for tumor growth recently proposed in citecwsl. 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 $vu$ satisfies $vucdotnu>0$, where $nu$ is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.

Appeared in

  • Nonlinearity, 30:4 (2017).

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