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We present a two-species model with applications in tumour modelling. Themain novelty is the coupling of both species through the so-called Brinkman lawwhich is typically used in the context of visco-elastic media, where thevelocity field is linked to the total population pressure via an ellipticequation. The same model for only one species has been studied by Perthame andVauchelet in the past. The first part of this paper is dedicated toestablishing existence of solutions to the problem, while the second part dealswith the incompressible limit as the stiffness of the pressure law tends toinfinity. Here we present a novel approach in one spatial dimension thatdiffers from the kinetic reformulation used in the aforementioned study and,instead, relies on uniform BV-estimates.

Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension