Several recent papers have addressed the modelling of tissue growth by multi-phase models where the velocity is related to the pressure by one of the physical laws (Stokes’, Brinkman’s or Darcy’s). While each of these models has been extensively studied, not so much is known about the connection between them. In the recent paper (David et al. in SIAM J. Math. Anal. 56(2):2090–2114, 2024), assuming the linear form of the pressure, the Authors connected two multi-phase models by an inviscid limit: the viscoelastic one (of Brinkman’s type) and the inviscid one (of Darcy’s type). Here, we prove that the same is true for a nonlinear, power-law pressure. The new ingredient is that we use the relation between the pressure p and the Brinkman potential W to deduce compactness in space of p from the compactness in space of W.

In 1934, Leray (Acta Math 63:193–248, 1934) proved the existence of global-in-time weak solutions to the Navier–Stokes equations for any divergence-free initial data in L2(R3). In the 1980s, Giga (J Differ Equ 62(2):186–212, 1986) and Kato (Math Z 187(4):471–480, 1984) independently showed that there exist global-in-time mild solutions corresponding to small enough critical L3(R3) initial data. In 1990, Calderón (Trans Am Math Soc 318:179–200, 1990) filled the gap to show that there exist global-in-time weak solutions for all supercritical initial data in Lp(R3) for 2<p<3 by utilising a splitting argument, blending the constructions of Leray and Giga-Kato. In this paper, we utilise a “Calderón-like” splitting to show the global-in-time existence of weak solutions to the Navier–Stokes equations corresponding to supercritical Besov space initial data u0∈B˙q,∞s(R3) where q>22$$\\end{document}]]> and -1+2q<s<min-1+3q,0, which fills a similar gap between Leray and known mild solution theory in the Besov space setting. We also use the Calderón-like splitting to investigate the structure of the singular set under a Type-I blow-up assumption in the Besov space setting, which is considerably rougher than in previous works.