Quantitative Control of Solutions to the Axisymmetric Navier-Stokes Equations in Terms of the Weak \(L^3\) Norm

We are concerned with strong axisymmetric solutions to the 3D incompressible Navier-Stokes equations. We show that if the weak \(L^3\) norm of a strong solution u on the time interval [0, T] is bounded by \(A \gg 1\) then for each \(k\ge 0 \) there exists \(C_k>1\) such that \(\Vert D^k u (t) \Vert _{L^\infty (\mathbb {R}^3)} \le t^{-(1+k)/2}\exp \exp A^{C_k}\) for all \(t\in (0,T]\).