Quantitative blow-up suppression for the Patlak-Keller-Segel(-Navier-Stokes) system via Couette flow on $\mathbb{R}^2$

It is well known that solutions to the Patlak--Keller--Segel system on $\mathbb{R}^2$ blow up in finite time if the initial mass exceeds $8\pi$. In this paper, we investigate the mixing effect induced by a Couette flow $(Ay, 0)$ with a quantitatively determined amplitude $A$, which suppresses bacterial aggregation. For the Patlak--Keller--Segel system advected by such a flow on $\mathbb{R}^2$, we prove that the solutions remain global in time even for large initial mass, provided the amplitude $A$ is sufficiently large. Specifically, global well-posedness holds if $A$ satisfies a lower bound of the form $C_* \left(\| \langle D_x\rangle^{m} \langle {D_x}^{-1}\rangle^\epsilon n_{\mathrm{in}} \|_{L^2}^2+1\right)^{9/2}$. A notable feature of our result is the explicit estimate of the sufficient constant, given by $C_* = 2,058,614$. Furthermore, for the coupled Patlak--Keller--Segel--Navier--Stokes system near the Couette flow, we establish an analogous global existence result, provided the amplitude is sufficiently large in form of $C_*\|(n_{\rm in}, |D_x|^{1/3} n_{\rm in},\omega_{\rm in})\|_{Y_{m,\epsilon}}^9$.