Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On regular periodic solutions to the Navier-Stokes equations   We find a global a priori estimate for solutions to the Navier-Stokes
equations with periodic boundary conditions guaranteeing in view of the Serrin
type condition the existence of global regular solutions. We derive the
following estimate $$ \lVert V(t) \rVert_{H^1(\Omega)}\leq c, \qquad (1) $$
where $V$ is the velocity of the fluid. The estimate (1) is proved in two
steps. First we derive a global estimate guaranteeing the existence of global
regular solutions to weakly compressible Navier-Stokes equations with large
second viscosity, density close to a constant and gradient part of velocity
small. Next we show that solutions to the Navier-Stokes equations remain close
to solutions to the weakly compressible Navier-Stokes equations if the
corresponding initial data and external forces are sufficiently close.

16
The categories of this paper are: stat
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ