Abstract
Abstract
In Liu and Zhang (2020 Arch. Ration. Mech. Anal.
235 1405–44); Liu et al (2020 Arch. Ration. Mech. Anal.
238 805–43), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the (anisotropic) Navier–Stokes (NS) system has a unique global solution. The goal of this paper is to extend this type of result to the 3D inhomogeneous (density-dependent) NS system. More precisely, given initial density such that
a
0
≜
1
ρ
0
−
1
∈
B
p
,
1
3
p
(
R
3
)
and the initial velocity
u
0
=
(
u
0
h
,
u
0
3
)
∈
B
p
,
1
−
1
+
2
p
,
1
p
(
R
3
)
,
with
u
0
h
belonging to
H
1
(
R
3
)
, then the inhomogeneous NS system has a unique global solution provided that
(
∥
a
0
∥
B
p
,
1
3
p
+
∥
Λ
h
−
1
∂
3
u
0
∥
B
p
,
1
−
1
+
2
p
,
1
p
)
⋅
f
(
u
0
)
being sufficiently small for some bounded function f depending on
∥
u
0
∥
p
,
1
−
1
+
2
p
,
1
p
and
∥
u
0
h
∥
H
1
. This provide a more general result that of Chemin et al (2014 Commun. Math. Phys.
272 529–66); Chemin and Zhang (2015 Commun. PDE
40 878–96).
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics