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 \[ \left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \Big(n(n+1)^{-\alpha}\nabla c\Big), c_t + u\cdot\nabla c &=& \Delta c-c+n, u_t +\nabla P &=& \Delta u + n \nabla \phi + f(x,t), \qquad \nabla\cdot u =0, \end{array} \right. \qquad \qquad (\star) \] is considered in a bounded domain $\Omega\subset R^3$, 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 \[ \alpha>\frac{1}{3}, \] 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.