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

Abstract

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 $C_{\ast }{\left(\Vert ⟨D_x{⟩}^m⟨{D_x}^{-1}{⟩}^{\epsilon }{n}_{\mathrm{in}}{\Vert }_{L^2}^2+1\right)}^{9/2}$. A notable feature of our result is the explicit estimate of the sufficient constant, given by $C_{\ast }=2,058,614$. 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 $C_{\ast }\Vert (n_{\mathrm{in}},|D_x{|}^{1/3}{n}_{\mathrm{in}},{\omega }_{\mathrm{in}}){\Vert }_{Y_{m,\epsilon }}^9$.