# On the vanishing viscosity approximation of a nonlinear model for tumor growth

## TL;DR

This paper studies a complex nonlinear model of tumor growth involving multiple cell types, nutrients, and drugs, using a vanishing viscosity approach to prove the existence of global weak solutions without symmetry or small data restrictions.

## Contribution

It introduces a novel application of vanishing viscosity methods to a multi-phase tumor growth model with evolving domain and no symmetry assumptions.

## Key findings

- Existence of global-in-time weak solutions for the model.
- The approach handles large initial data and general domains.
- No symmetry assumptions are required.

## Abstract

We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Darcy's equation. The domain occupied by the tumor in this setting is a growing continuum $\Omega$ with boundary $\partial \Omega$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on the vanishing viscosity of the Brinkman's regularization. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data.