Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lower bound on the blow-up rate of the axisymmetric Navier-Stokes
  equations   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 $0<C_*<\infty$
allowed to be large, we then prove that $v$ is regular at time zero.

37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics