  We establish the vanishing viscosity limit of the Navier-Stokes equations to
the isentropic Euler equations for one-dimensional compressible fluid flow. For
the Navier-Stokes equations, there exist no natural invariant regions for the
equations with the real physical viscosity term so that the uniform sup-norm of
solutions with respect to the physical viscosity coefficient may not be
directly controllable and, furthermore, convex entropy-entropy flux pairs may
not produce signed entropy dissipation measures. To overcome these
difficulties, we first develop uniform energy-type estimates with respect to
the viscosity coefficient for the solutions of the Navier-Stokes equations and
establish the existence of measure-valued solutions of the isentropic Euler
equations generated by the Navier-Stokes equations. Based on the uniform
energy-type estimates and the features of the isentropic Euler equations, we
establish that the entropy dissipation measures of the solutions of the
Navier-Stokes equations for weak entropy-entropy flux pairs, generated by
compactly supported $C^2$ test functions, are confined in a compact set in
$H^{-1}$, which lead to the existence of measure-valued solutions that are
confined by the Tartar-Murat commutator relation. A careful characterization of
the unbounded support of the measure-valued solution confined by the commutator
relation yields the reduction of the measure-valued solution to a Delta mass,
which leads to the convergence of solutions of the Navier-Stokes equations to a
finite-energy entropy solution of the isentropic Euler equations.
