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

We study the incompressible limit for a two-species model with applications to tissue growth in the case of coupling through the so-called Brinkman's law in any space dimensions. The coupling through this elliptic equation accounts for viscosity effects among the individual species. In a recent paper Dębiec & Schmidtchen established said result in one spacial dimension, with their proof hinging on being able to establish uniform BV-bounds. This approach is fundamentally different from the one-species case in arbitrary dimension, established by Perthame & Vauchelet. Their result relies on a kinetic reformulation to obtain strong compactness of the pressure. In this paper we fill this gap in the literature and present the incompressible limit for the system in arbitrary space dimension. The difficulty stems from jump discontinuities in the pressure not only at the boundary of the support of the two species but also at internal layers giving rise to the question as to how compactness can be obtained. The answer is a combination of techniques consisting of the application of the compactness method of Bresch & Jabin, an adaptation of the aforementioned kinetic reformulation, and several parallels to the one dimensional strategy. The main result of this paper establishes a rigorous bridge between the population dynamics of growing tissue at a density level and a geometric model thereof.