Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

We revisit the regularity theory of Escauriaza, Seregin, and \v{S}ver\'ak for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(\mathbf{R}^3)$ 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 $L^3_x$ bound (with a dependence that is triple exponential in nature). In particular, we show that as one approaches a finite blowup time $T_*$, the critical $L^3_x$ norm must blow up at a rate $(\log\log\log \frac{1}{T_*-t})^c$ or faster for an infinite sequence of times approaching $T_*$ and some absolute constant $c>0$.