Incompressible limit of mechanical model of tumor growth with viscosity

Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two levels of description considering the ‘stiff pressure law’ limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not imply immediately compactness. For instance, traveling wave solutions and numerical simulations show that the pressure may be discontinous in space which is not the case for the elastic case. Key-words: Tumor growth; Hele-Shaw equation; Free boundary problems; Porous media; Viscoelastic media Mathematical Classification numbers: 35J60; 35K57; 74J30; 92C10; 1 The cell model with visco-elastic flow We consider a mechanical model of tumor growth considered as a visco-elastic media. We denote the number density of tumor cells by n(x, t), the pressure by p(x, t) and we assume a Brinkman flow that means the macroscopic velocity field is given by ∇W for a potential W closely related to the pressure. With these assumptions, the model for tumor growth writes ∂tnk − div(nk∇Wk) = nkG (