Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?

Abstract

A class of Keller–Segel–Stokes systems generalizing the prototype

$$\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot \left( n(n+1)^{-\alpha }\nabla c\right) , \\ c_t + u\cdot \nabla c = \Delta c-c+n, \\ u_t +\nabla P = \Delta u + n \nabla \phi + f(x,t), \quad \nabla \cdot u =0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$

is considered in a bounded domain $\Omega \subset \mathbb {R}^3$, where $\phi$ and f are given sufficiently smooth functions such that f is bounded in $\Omega \times (0,\infty )$. It is shown that under the condition that

$$\begin{aligned} \alpha >\frac{1}{3}, \end{aligned}$$

for all sufficiently regular initial data a corresponding Neumann–Neumann–Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption $\alpha >\frac{1}{2}$. In view of known results on the existence of exploding solutions when $\alpha <\frac{1}{3}$, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $\frac{1}{3}$, as previously found for the fluid-free counterpart of ($\star$), remains essentially unaffected by fluid interaction of the type considered here.