Abstract
We prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations
is regular in
(
0
,
T
]
{(0,T]}
if the
L
1
(
0
,
T
;
B
˙
∞
,
∞
-
1
)
{L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})}
-
or
L
2
(
0
,
T
;
B
˙
∞
,
∞
-
1
)
{L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})}
-norm of
u
k
/
2
,
k
{u_{k/2,k}}
, the mid frequency part of Fourier modes
k
2
≤
|
ξ
|
<
k
{\frac{k}{2}\leq|\xi|<k}
,
is small depending on the kinematic viscosity ν, initial value
u
0
{u_{0}}
and
the maximum of an averaged energy
dissipation rate
A
≡
sup
t
∈
(
0
,
T
)
(
ν
t
-
1
∫
0
t
∥
∇
u
∥
2
𝑑
τ
)
A\equiv\sup_{t\in(0,T)}\biggl{(}\nu t^{-1}\int_{0}^{t}\lVert\nabla u\rVert^{2}%
\,d\tau\biggr{)}
for some
k
≥
k
0
(
ν
,
u
0
,
A
)
{k\geq k_{0}(\nu,u_{0},A)}
. In particular, when a sufficiently high frequency part of
u
0
{u_{0}}
decays fast at an exponential rate, then we obtain regularity
conditions in terms of smallness of the
L
1
(
0
,
T
;
B
˙
∞
,
∞
-
1
)
{L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})}
-
or
L
2
(
0
,
T
;
B
˙
∞
,
∞
-
1
)
{L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})}
-norm of
u
k
/
2
,
k
{u_{k/2,k}}
, which involve only the known data ν and
u
0
{u_{0}}
.
Frontmatter
