The Stokes Limit in a Three-Dimensional Keller–Segel–Navier–Stokes System

Abstract

In this paper, we consider the initial-boundary value problem of Keller–Segel–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{aligned}&n_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla n^{\kappa } =\varDelta n^{\kappa } - \nabla \cdot \big (n^{\kappa }(1+n^{\kappa })^{-\alpha }\nabla c^{\kappa } \big ),&\qquad x\in \varOmega ,\,t>0, \\&c_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla c^{\kappa } =\varDelta c^{\kappa } -c^{\kappa } +n^{\kappa },&\qquad x\in \varOmega ,\,t>0,\\&\mathbf{u}^{\kappa }_t+\kappa ({\mathbf {u}}^{\kappa } \cdot \nabla )\mathbf{u}^{\kappa } =\varDelta \mathbf{u}^{\kappa } -\nabla P^{\kappa } + n^{\kappa }\nabla \phi ,&\qquad x\in \varOmega ,\,t>0,\\&\nabla \cdot \mathbf{u}^{\kappa } =0,&\qquad x\in \varOmega ,\,t>0\\ \end{aligned} \right. \end{aligned}$$

with no-flux boundary conditions for $n^{\kappa }$ and $c^{\kappa }$ and a no-slip boundary condition for $\mathbf{u}^{\kappa }$, and with sufficiently regular initial data in a bounded domain $\varOmega \subset {\mathbb {R}}^3\,$ with smooth boundary. Our result reveals that the solution $(n^{\kappa },\,c^{\kappa },\,\mathbf{} u^{\kappa })$ converges towards the solution of the corresponding Stokes variant $(\kappa =0)$ with an exponential time decay rate as $\kappa \rightarrow 0_{+}$ provided that $\Vert n_0\Vert _{L^\frac{3}{2}(\varOmega )},$ $\Vert \nabla c_0\Vert _{L^3(\varOmega )}$ and $\Vert {\mathbf {u}}_0\Vert _{L^3(\varOmega )}$ are suitable small. Therefore, we can extend the obtained result on the Stokes limit in the two-dimensional case to the more complicated three-dimensional case.