Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2021-07-15 , DOI: 10.1007/s10884-021-10043-z Ju Zhou 1
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.
中文翻译：
三维 Keller-Segel-Navier-Stokes 系统中的斯托克斯极限
在本文中，我们考虑了 Keller-Segel-Navier-Stokes 系统的初边值问题
$$\begin{aligned} \left\{ \begin{aligned}&n_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla n^{\kappa } =\varDelta n^{\ kappa } - \nabla \cdot \big (n^{\kappa }(1+n^{\kappa })^{-\alpha }\nabla c^{\kappa } \big ),&\qquad x\in \varOmega ,\,t>0, \\&c_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla c^{\kappa } =\varDelta c^{\kappa } -c^ {\kappa } +n^{\kappa },&\qquad x\in \varOmega ,\,t>0,\\&\mathbf{u}^{\kappa }_t+\kappa ({\mathbf {u} }^{\kappa } \cdot \nabla )\mathbf{u}^{\kappa } =\varDelta \mathbf{u}^{\kappa } -\nabla P^{\kappa } + n^{\kappa } \nabla \phi ,&\qquad x\in \varOmega ,\,t>0,\\&\nabla \cdot \mathbf{u}^{\kappa } =0,&\qquad x\in \varOmega ,\ ,t>0\\ \end{aligned} \right。\end{对齐}$$\(n^{\kappa }\)和\(c^{\kappa }\)的无通量边界条件和\(\mathbf{u}^{\kappa }\)的无滑移边界条件，并且在边界平滑的有界域\(\varOmega \subset {\mathbb {R}}^3\,\) 中具有足够规则的初始数据。我们的结果表明，解\((n^{\kappa },\,c^{\kappa },\,\mathbf{} u^{\kappa })\)收敛于相应的 Stokes 变体\ ((\kappa =0)\)的指数时间衰减率为\(\kappa \rightarrow 0_{+}\)条件是\(\Vert n_0\Vert _{L^\frac{3}{2}( \varOmega )},\) \(\Vert \nabla c_0\Vert _{L^3(\varOmega )}\)和\(\Vert {\mathbf {u}}_0\Vert _{L^3(\varOmega )}\)适合小。因此，我们可以将得到的二维情况下斯托克斯极限的结果推广到更复杂的三维情况。
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