Suppression of blow-up in multispecies Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow in a finite channel (Q6652058)

In this paper, the authors considerED the multispecies parabolic-elliptic Patlak-Keller-Segelsystem coupled with the Navier-Stokes equations near the two-dimensional Poiseuille flow \((A(1-y2),0)\) in a finite channel \(\mathbb T \times \mathbb I\) with \(\mathbb I = [- 1,1]\). Furthermore, the Navier-slip boundary condition is imposed on the perturbation of velocity \(u\). They showed that if \(A\) is large enough, the solutions to the system are global in time without any smallness restriction on the initial cell mass or the initial velocity. Compared with the Couette flow, we used the resolvent estimate of the Orr-Sommerfeld equation introduced by \textit{T. Li} et al. [Commun. Pure Appl. Math. 73, No. 3, 465--557 (2020; Zbl 1442.35346)]. This seems to be the first result of considering boundary effects when studying the suppression effect of the shear flow in the Patlak-Keller-Segel-Navier-Stokes model.\N\NThese results are undoubtedly novel and intriguing, and they extend previous understanding in a seamless manner. The derivation is presented and organized exceptionally well, building upon existing ideas from the literature and appropriately citing the relevant references. Moreover, the work introduces significant and nontrivial new ideas. Overall, in my opinion, this is an exemplary piece of research.