AbstractThe flux-limited Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
n
t
+
u
·
∇
n
=
Δ
n
-
∇
·
(
n
f
(
|
∇
c
|
2
)
∇
c
)
,
c
t
+
u
·
∇
c
=
Δ
c
-
c
+
n
,
u
t
+
(
u
·
∇
)
u
=
Δ
u
+
∇
P
+
n
∇
Φ
,
∇
·
u
=
0
,
(
⋆
)
is considered in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^2$$
Ω
⊂
R
2
. It is shown that whenever the suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that $$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$
|
f
(
ξ
)
|
≤
K
f
·
(
ξ
+
1
)
-
α
2
holds for all $$\xi \ge 0$$
ξ
≥
0
with some $$K_f>0$$
K
f
>
0
and $$\alpha >0$$
α
>
0
, for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in $$L^\infty (\Omega \times (0,\infty ))$$
L
∞
(
Ω
×
(
0
,
∞
)
)
with respect to all its components. By extending a corresponding result known for a fluid-free counterpart of ($$\star $$
⋆
), this confirms that with regard to the possible emergence of blow-up phenomena, the choice $$f\equiv const.$$
f
≡
c
o
n
s
t
.
retains some criticality also in the presence of fluid interaction.
Finite-Time Blow-up in a Quasilinear Degenerate Chemotaxis System with Flux Limitation
