Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Cauchy problem for the nonlinear viscous Boussinesq equation in the
  $L^q$ framework   In this paper, we study the viscous Boussinesq equation in the whole space
$\mathbb{R}^n$, which describes the propagation of small amplitude and long
waves on the surface of water with viscous effects. Concerning the linearized
Cauchy problem, some qualitative properties of solutions including $L^m-L^q$
estimates with $1\leqslant m\leqslant q\leqslant \infty$, inviscid limits and
asymptotic profiles of solution with respect to the small viscosity are
investigated by means of the Fourier analysis and the WKB method. For another,
by applying some fractional order interpolations in the harmonic analysis, we
derive the $L^q$ well-posedness and estimates for small data solutions to the
nonlinear viscous Boussinesq equation under some conditions for the parameter
of nonlinearity.

23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th