We revisit the regularity theory of Escauriaza, Seregin, and Šverák for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical
L
x
3
(
R
3
)
L^3_x(\mathbb {R}^3)
norm. By replacing all invocations of compactness methods in these arguments with quantitative substitutes, and similarly replacing unique continuation and backwards uniqueness estimates by their corresponding Carleman inequalities, we obtain quantitative bounds for higher regularity norms of these solutions in terms of the critical
L
x
3
L^3_x
bound (with a dependence that is triple exponential in nature). In particular, we show that as one approaches a finite blowup time
T
∗
T_*
, the critical
L
x
3
L^3_x
norm must blow up at a rate
(
log
log
log
1
T
∗
−
t
)
c
(\log \log \log \frac {1}{T_*-t})^c
or faster for an infinite sequence of times approaching
T
∗
T_*
and some absolute constant
c
>
0
c>0
.