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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. Another longstanding problem is whether a rigorous
proof could be provided for the inviscid limit of the multidimensional
compressible Navier-Stokes to Euler equations with large initial data. 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 even when the total initial-energy is unbounded.Comment: 57 page
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