Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations

Abstract

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in ${\mathbb{R}}^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v(x,t)|\le C_{\ast }(r^2-t{)}^{-1/2}$ for $-T_0\le t<0$ and $0<C_{\ast }<\infty$ allowed to be large, we then prove that $v$ is regular at time zero.

25 pages

References in corpus (1)

• Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations

Cited by in corpus (6)

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• Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions