Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vanishing of vacuum states and blow-up phenomena of the compressible
  Navier-Stokes equations   The Navier-Stokes systems for compressible fluids with density-dependent
viscosities are considered in the present paper. These equations, in
particular, include the ones which are rigorously derived recently as the
Saint-Venant system for the motion of shallow water, from the Navier-Stokes
system for incompressible flows with a moving free surface [14]. These
compressible systems are degenerate when vacuum state appears. We study
initial-boundary-value problems for such systems for both bounded spatial
domains or periodic domains. The dynamics of weak solutions and vacuum states
are investigated rigorously.
  First, it is proved that the entropy weak solutions for general large initial
data satisfying finite initial entropy exist globally in time. Next, for more
regular initial data, there is a global entropy weak solution which is unique
and regular with well-defined velocity field for short time, and the interface
of initial vacuum propagates along particle path during this time period. Then,
it is shown that for any global entropy weak solution, any (possibly existing)
vacuum state must vanish within finite time. The velocity (even if regular
enough and well-defined) blows up in finite time as the vacuum states vanish.
Furthermore, after the vanishing of vacuum states, the global entropy weak
solution becomes a strong solution and tends to the non-vacuum equilibrium
state exponentially in time.

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The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio