Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global-in-time solutions to Navier-Stokes equations without external
  force   Let $\Omega$ be either $\mathbb{R}^3$ or a bounded domain in $\mathbb{R}^3$.
The incompressible Navier-Stokes equations in $\Omega$ without external force
is considered:
  $\partial _tu-\nu\Delta u+ (u\cdot\nabla)u+\nabla p=0$, $u(.,0)=u_0$ and
$div~u=0$.
  I prove the non-existence of blow-up-in-finite-time strong solution, and then
imply the existence of strong solutions on the time interval $[0,+\infty)$.

30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
3
The categories of this paper are: physics
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat