First, we show that if the pressure
p
p
(associated to a weak Leray–Hopf solution
v
v
of the Navier–Stokes equations) satisfies
‖
p
‖
L
t
∞
(
0
,
T
∗
;
L
3
2
,
∞
(
R
3
)
)
≤
M
2
,
\begin{equation*} \|p\|_{L^{\infty }_{t}(0,T^*; L^{\frac {3}{2},\infty }(\mathbb {R}^3))}\leq M^2, \end{equation*}
then
v
v
possesses higher integrability up to the first potential blow-up time
T
∗
T^*
. Our method is concise and is based upon energy estimates applied to powers of
|
v
|
|v|
and the utilization of a “small exponent”.
As a consequence, we show that if a weak Leray–Hopf solution
v
v
first blows up at
T
∗
T^*
and satisfies the Type I condition
‖
v
‖
L
t
∞
(
0
,
T
∗
;
L
3
,
∞
(
R
3
)
)
≤
M
\|v\|_{L^{\infty }_{t}(0,T^*; L^{3,\infty }(\mathbb {R}^3))}\leq M
, then
∇
v
∈
L
2
+
O
(
1
M
)
(
R
3
×
(
1
2
T
∗
,
T
∗
)
)
.
\begin{equation*} \nabla v\in L^{2+O(\frac {1}{M})}(\mathbb {R}^3\times (\tfrac {1}{2}T^*,T^*)). \end{equation*}
This is the first result of its kind, improving the integrability exponent of
∇
v
\nabla v
under the Type I assumption in the three-dimensional setting.
Finally, we show that if
v
:
R
3
×
[
−
1
,
0
]
→
R
3
v:\mathbb {R}^3\times [-1,0]\rightarrow \mathbb {R}^3
is a weak Leray–Hopf solution to the Navier–Stokes equations with
s
n
↑
0
s_{n}\uparrow 0
such that
sup
n
‖
v
(
⋅
,
s
n
)
‖
L
3
,
∞
(
R
3
)
≤
M
\begin{equation*} \sup _{n}\|v(\cdot ,s_{n})\|_{L^{3,\infty }(\mathbb {R}^3)}\leq M \end{equation*}
then
v
v
possesses at most
O
(
M
20
)
O(M^{20})
singular points at
t
=
0
t=0
. Our method is direct and concise. It is based upon known
ε
\varepsilon
-regularity, global bounds on a Navier–Stokes solution with initial data in
L
3
,
∞
(
R
3
)
L^{3,\infty }(\mathbb {R}^3)
and rescaling arguments. We do not require arguments based on backward uniqueness nor unique continuation results for parabolic operators.