  We prove the existence of relative finite-energy vanishing viscosity
solutions of the one-dimensional, isentropic Euler equations under the
assumption of an asymptotically isothermal pressure law, that is, $p(\rho)/\rho
= O(1)$ in the limit $\rho \to \infty$. This solution is obtained as the
vanishing viscosity limit of classical solutions of the one-dimensional,
isentropic, compressible Navier--Stokes equations. Our approach relies on the
method of compensated compactness to pass to the limit rigorously in the
nonlinear terms. Key to our strategy is the derivation of hyperbolic
representation formulas for the entropy kernel and related quantities; among
others, a special entropy pair used to obtain higher uniform integrability
estimates on the approximate solutions. Intricate bounding procedures relying
on these representation formulas then yield the required compactness of the
entropy dissipation measures. In turn, we prove that the Young measure
generated by the classical solutions of the Navier--Stokes equations reduces to
a Dirac mass, from which we deduce the required convergence to a solution of
the Euler equations.
