Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global Solutions of the Compressible Euler Equations with Large Initial
  Data of Spherical Symmetry and Positive Far-Field Density   We are concerned with the global existence theory for spherically symmetric
solutions of the multidimensional compressible Euler equations with large
initial data of positive far-field density. The central feature of the
solutions is the strengthening of waves as they move radially inward toward the
origin. Various examples have shown that the spherically symmetric solutions of
the Euler equations blow up near the origin at certain time. A fundamental
unsolved problem is whether the density of the global solution would form
concentration to become a measure near the origin for the case when the total
initial-energy is unbounded and the wave propagation is not at a finite speed
starting initially. In this paper, we establish a global existence theory for
spherically symmetric solutions of the compressible Euler equations with large
initial data of positive far-field density and relative finite-energy. This is
achieved by developing a new approach via adapting a class of degenerate
density-dependent viscosity terms, so that a rigorous proof of the vanishing
viscosity limit of global weak solutions of the Navier-Stokes equations with
the density-dependent viscosity terms to the corresponding global solution of
the Euler equations with large initial data of spherical symmetry and positive
far-field density can be obtained. One of our main observations is that the
adapted class of degenerate density-dependent viscosity terms not only includes
the viscosity terms for the Navier-Stokes equations for shallow water (Saint
Venant) flows but also, more importantly, is suitable to achieve our key
objective of this paper. These results indicate that concentration is not
formed in the vanishing viscosity limit for the Navier-Stokes approximations
constructed in this paper even when the total initial-energy is unbounded.

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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol