In this paper, we consider the linear stability of blowup solution for incompressible Keller–Segel–Navier–Stokes system in whole space $\mathbb{R}^{3}$ . More precisely, we show that, if the initial data of the three dimensional Keller–Segel–Navier–Stokes system is close to the smooth initial function $(0,0,\textbf{u}_{s}(0,x) )^{T}$ , then there exists a blowup solution of the three dimensional linear Keller–Segel–Navier–Stokes system satisfying the decomposition $$ \bigl(n(t,x),c(t,x),\textbf{u}(t,x) \bigr)^{T}= \bigl(0,0, \textbf{u}_{s}(t,x) \bigr)^{T}+\mathcal{O}(\varepsilon ), \quad \forall (t,x)\in \bigl(0,T^{*}\bigr) \times \mathbb{R}^{3}, $$ in Sobolev space $H^{s}(\mathbb{R}^{3})$ with $s=\frac{3}{2}-5a$ and constant $0< a\ll 1$ , where $T^{*}$ is the maximal existence time, and $\textbf{u}_{s}(t,x)$ given in (Yan 2018) is the explicit blowup solution admitted smooth initial data for three dimensional incompressible Navier–Stokes equations.

中文翻译：

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不可压缩的Keller-Segel-Navier-Stokes系统的爆破溶液的线性稳定性

在本文中，我们考虑了不可压缩的Keller–Segel–Navier–Stokes系统在整个空间$ \ mathbb {R} ^ {3} $中的爆破解的线性稳定性。更准确地说，我们表明，如果三维Keller-Segel-Navier-Stokes系统的初始数据接近平滑初始函数$（0,0，\ textbf {u} _ {s}（0，x） ）^ {T} $，则存在满足分解$$ \ bigl（n（t，x），c（t，x），\ textbf的三维线性Keller-Segel-Navier-Stokes系统的分解解决方案{u}（t，x）\ bigr）^ {T} = \ bigl（0,0，\ textbf {u} _ {s}（t，x）\ bigr）^ {T} + \ mathcal {O} （\ varepsilon），\ quad \ forall（t，x）\ in \ bigl（0，T ^ {*} \ bigr）\ times \ mathbb {R} ^ {3}，Sobolev空间$ H ^ { s}（\ mathbb {R} ^ {3}）$与$ s = \ frac {3} {2} -5a $和常数$ 0 <a \ ll 1 $，其中$ T ^ {*} $是最大值存在时间和$ \ textbf {u} _ {s}（t，