  We are concerned with spherically symmetric solutions of the Euler equations
for multidimensional compressible fluids, which are motivated by many important
physical situations. Various evidences indicate that spherically symmetric
solutions of the compressible Euler equations may blow up near the origin at
certain time under some circumstance. The central feature is the strengthening
of waves as they move radially inward. A longstanding open, fundamental
question is whether concentration could form at the origin. In this paper, we
develop a method of vanishing viscosity and related estimate techniques for
viscosity approximate solutions, and establish the convergence of the
approximate solutions to a global finite-energy entropy solution of the
compressible Euler equations with spherical symmetry and large initial data.
This indicates that concentration does not form in the vanishing viscosity
limit, even though the density may blow up at certain time. To achieve this, we
first construct global smooth solutions of appropriate initial-boundary value
problems for the Euler equations with designed viscosity terms, an approximate
pressure function, and boundary conditions, and then we establish the strong
convergence of the viscosity approximate solutions to a finite-energy entropy
solutions of the Euler equations.
