Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence of weak solutions to the three-dimensional steady
  compressible Navier-Stokes equations in bounded domains   We prove the existence of a weak solution to the three-dimensional steady
compressible isentropic Navier-Stokes equations in bounded domains for any
specific heat ratio \gamma > 1. Generally speaking, the proof is based on the
new weighted estimates of both pressure and kinetic energy for the approximate
system which result in some higher integrability of the density, and the method
of weak convergence. Comparing with [12] where the spatially periodic case was
studied, here we have to control the additional integral terms of both pressure
and kinetic energy involving with the points near the boundary which become
degenerate when the points approach the boundary. Such integral terms are
estimated using some new techniques, i.e., we use the techniques of the mirror
image and boundary straightening to prove that the weighted estimates of both
pressure and kinetic energy for the points near the boundary can be controlled
by the weighted estimates for the points on the boundary. Moreover, we prove
that once the weighted estimates of the kinetic energy in the direction of the
unit normal to the boundary are bounded, we can control the weighted estimates
of the total energy on the boundary.

26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th