A degenerate cross-diffusion system as the inviscid limit of a nonlocal tissue growth model

In recent years, there has been a spike in interest in multiphase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law, or Darcy's law. While each of these velocity -pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth of results in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumor model (of Brinkman type) and an inviscid tumor model (of Darcy type). Specifically, we prove the convergence of solutions of the Brinkman nonlocal transport system toward a weak solution of the Darcy nonlinear parabolic system in the limit of vanishing viscosity.