Does repulsion-type directional preference in chemotactic migration continue to regularize Keller–Segel systems when coupled to the Navier–Stokes equations?

Abstract

The repulsive Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{llll} n_t + u\cdot \nabla n &{}=&{} \Delta n + \nabla \cdot (n\nabla c), &{}\quad x\in \Omega , \; t>0, \\ c_t + u\cdot \nabla c &{}=&{} \Delta c -c+n, &{}\quad x\in \Omega , \; t>0, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \quad \nabla \cdot u=0, &{}\quad x\in \Omega , \; t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$is considered in smoothly bounded planar domains, where $$\Phi \in W^{2,\infty }(\Omega )$$ is given. It is well-known that the corresponding fluid-free analogue, when posed under homogeneous no-flux boundary conditions, admits global classical solutions for arbitrarily large initial data, thus substantially differing from the classical two-dimensional Keller–Segel system featuring chemoattraction-driven finite-time blow-up for some initial data. The literature on such chemorepulsion systems, however, strongly relies on the presence of an associated energy structure which is apparently destroyed by the fluid interaction mechanism in ($$\star $$). By making use of appropriate functional inequalities involving certain logarithmic expressions arising due to the planarity of the considered setting, it is shown that nevertheless an initial-boundary value problem for ($$\star $$) admits globally defined classical solutions for all reasonably regular initial data.