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