UCLA

Consider a velocity field $u$ solving the incompressible Navier-Stokes equations on $[0,T]\times\mathbb R^d$ ($d\geq3$) and satisfying $\|u(t)\|_X\leq A$ for all times, where the norm $X$ is critical with respect to the Navier-Stokes scaling. We prove several theorems to the effect that the regularity of the solution can be controlled explicitly in terms of $A$, building upon Tao's pioneering work on the case $d=3$, $X=L^3(\mathbb R^3)$. First we prove a generalization to the critical Lebesgue space in any number of spatial dimensions ($d\geq4$, $X=L^d(\mathbb R^d)$). Then we show a variety of circumstances under which Tao's bounds can be strengthened, including the case in which the solution is nearly axisymmetric. For exactly axisymmetric solutions, we prove regularity in terms of the weak norm $X=L^{3,\infty}(\mathbb R^3)$ which implies effective bounds on approximately self-similar behavior.