Boundedness in a two-dimensional Keller–Segel–Navier–Stokes system involving a rapidly diffusing repulsive signal

Abstract

This paper is concerned with the Keller–Segel–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot \nabla n &{}=&{} \Delta n + \nabla \cdot (n\nabla c), \qquad &{} x\in \Omega , \ t>0, \\ u\cdot \nabla c &{}=&{} \Delta c -c+n, \qquad &{} x\in \Omega , \ t>0, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$

with a given smooth gravitational potential $\Phi$. It is shown that for all suitably regular initial data, a corresponding no-flux/no-flux/Dirichlet initial boundary value problem posed in a smoothly bounded planar domain admits a uniquely determined global classical solution (n, c, u, P) which has the additional property that n remains uniformly bounded. This partially goes beyond a recent result asserting global classical solvability, but without any boundedness information, in a related slightly more complex variant of ($\star$) accounting for parabolic evolution of the quantity c. In particular, the obtained outcome therefore provides further evidence indicating that the considered fluid interaction does not substantially reduce a certain explosion-avoiding character of the Keller–Segel-type chemorepulsion mechanisms, as known to form an essential feature of corresponding fluid-free analogues. The reasoning at its core relies on the use of a quasi-Lyapunov inequality which operates at regularity levels that seem rather unusual in this and related contexts, but which in the considered two-dimensional setting can be seen to serve as a starting point sufficient for a bootstrap-type series of arguments, finally providing global boundedness.