Joint inviscid, incompressible, and continuous phenotype limit in nonlocal models of tissue growth

Abstract

We rigorously derive the joint limit of vanishing viscosity, singular pressure, and discrete-to-continuous phenotype structure in a class of tissue growth models. Starting from a viscoelastic (Brinkman-type) system describing multiple interacting phenotypes, where pressure depends nonlinearly on total cell density, we establish convergence to an incompressible Darcy-type model with a continuous phenotypic structure. Our analysis builds upon recent advances on the Brinkman-to-Darcy limit and incompressible transitions by David, Jacob, and Kim [arXiv:2503.18870], and on hydrodynamic limits for phenotype-structured populations by Debiec, Mandal, and Schmidtchen [J. Differential Equations 2025]. A key novelty lies in combining all three singular limits simultaneously, under uniform a priori estimates, compactness in space-phenotype-time, and a generalized entropy-dissipation structure. This provides a unified framework for modeling constrained tissue growth with mechanical feedback and phenotypic plasticity.