AbstractThis paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound $$\Vert u\Vert _{L^{\infty }_{t}L^{3,\infty }_{x}}\le M$$
‖
u
‖
L
t
∞
L
x
3
,
∞
≤
M
. Namely, we show that if $$T^*$$
T
∗
is a first blow-up time and $$(0,T^*)$$
(
0
,
T
∗
)
is a singular point then $$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{0}(R))}\ge C(M)\log \Big (\frac{1}{T^*-t}\Big ),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}). \end{aligned}$$
‖
u
(
·
,
t
)
‖
L
3
(
B
0
(
R
)
)
≥
C
(
M
)
log
(
1
T
∗
-
t
)
,
R
=
O
(
(
T
∗
-
t
)
1
2
-
)
.
We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on $${\mathbb {R}}^3\times (0,1)$$
R
3
×
(
0
,
1
)
with $$\begin{aligned} \sup _{n}\Vert u(\cdot ,t_{(n)})\Vert _{L^{3}({\mathbb {R}}^3)}<\infty \,\,\,\text {and}\,\,\,t_{(n)}\uparrow 1, \end{aligned}$$
sup
n
‖
u
(
·
,
t
(
n
)
)
‖
L
3
(
R
3
)
<
∞
and
t
(
n
)
↑
1
,
then u does not blow-up at $$t=1$$
t
=
1
. To prove our results we develop a new strategy for proving quantitative bounds for the Navier–Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.
Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration