Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weak and strong solutions of the $3D$ Navier-Stokes equations and their
  relation to a chessboard of convergent inverse length scales   Using the scale invariance of the Navier-Stokes equations to define
appropriate space-and-time-averaged inverse length scales associated with weak
solutions of the $3D$ Navier-Stokes equations, an infinite `chessboard' of
estimates for these inverse length scales is displayed in terms of labels
$(n,\,m)$ corresponding to $n$ derivatives of the velocity field in $L^{2m}$.
The $(1,\,1)$ position corresponds to the inverse Kolmogorov length $Re^{3/4}$.
These estimates ultimately converge to a finite limit, $Re^3$, as $n,\,m\to
\infty$, although this limit is too large to lie within the physical validity
of the equations for realistically large Reynolds numbers. Moreover, all the
known time-averaged estimates for weak solutions can be rolled into one single
estimate, labelled by $(n,\,m)$. In contrast, those required for strong
solutions to exist can be written in another single estimate, also labelled by
$(n,\,m)$, the only difference being a factor of 2 in the exponent. This
appears to be a generalisation of the Prodi-Serrin conditions for $n\geq 1$.

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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs