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

Abstract

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