Higher integrability and the number of singular points for the Navier–Stokes equations with a scale-invariant bound
Authors: Barker Tobias 
Published: September 2024

First, we show that if the pressure

p
p

(associated to a weak Leray–Hopf solution

v
v

of the Navier–Stokes equations) satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar p double-vertical-bar Subscript upper L Sub Subscript t Sub Superscript normal infinity Subscript left-parenthesis 0 comma upper T Sub Superscript asterisk Subscript semicolon upper L Sub Superscript three halves comma normal infinity Subscript left-parenthesis double-struck upper R cubed right-parenthesis right-parenthesis Baseline less-than-or-equal-to upper M squared comma">

‖
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar v double-vertical-bar Subscript upper L Sub Subscript t Sub Superscript normal infinity Subscript left-parenthesis 0 comma upper T Sub Superscript asterisk Subscript semicolon upper L Sub Superscript 3 comma normal infinity Subscript left-parenthesis double-struck upper R cubed right-parenthesis right-parenthesis Baseline less-than-or-equal-to upper M">

‖
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nabla v element-of upper L Superscript 2 plus upper O left-parenthesis StartFraction 1 Over upper M EndFraction right-parenthesis Baseline left-parenthesis double-struck upper R cubed times left-parenthesis one half upper T Superscript asterisk Baseline comma upper T Superscript asterisk Baseline right-parenthesis right-parenthesis period">

∇
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript n Endscripts double-vertical-bar v left-parenthesis dot comma s Subscript n Baseline right-parenthesis double-vertical-bar Subscript upper L Sub Superscript 3 comma normal infinity Subscript left-parenthesis double-struck upper R cubed right-parenthesis Baseline less-than-or-equal-to upper M">

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.