Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the initial value problem for the Navier-Stokes equations with the
  initial datum in the Sobolev spaces   In this paper, we study local well-posedness for the Navier-Stokes equations
with arbitrary initial data in homogeneous Sobolev spaces
$\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\
\frac{d}{p} - 1 \leq s < \frac{d}{2p}$. The obtained result improves the known
ones for $p > d$ and $s = 0$ (see M. Cannone (1995), M. Cannone and Y. Meyer
(1995)). In the case of critical indexes $s=\frac{d}{p}-1$, we prove global
well-posedness for Navier-Stokes equations when the norm of the initial value
is small enough. This result is a generalization of the ones in Cannone (1999)
and P. G. Lemarie-Rieusset (2002) in which $(p = d, s = 0)$ and $(p > d, s =
\frac{d}{p} - 1)$, respectively.

7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex