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

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

2007-01-27

arxiv.org/abs/math/0701796v1

International Mathematics Research Notices, Vol. 2008

Mathematics

Analysis of PDEs

25 pages

Scientific paper

10.1093/imrn/rnn016

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ 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_*{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and $0

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