  We study the Cauchy problem associated with the system of two conservation
laws arising in isothermal gas dynamics, in which the pressure and the density
are related by the $\gamma$-law equation $p(\rho) \sim \rho^\gamma$ with
$\gamma =1$. Our results complete those obtained earlier for $\gamma >1$. We
prove the global existence and compactness of entropy solutions generated by
the vanishing viscosity method. The proof relies on compensated compactness
arguments and symmetry group analysis. Interestingly, we make use here of the
fact that the isothermal gas dynamics system is invariant modulo a linear
scaling of the density. This property enables us to reduce our problem to that
with a small initial density. One symmetry group associated with the linear
hyperbolic equations describing all entropies of the Euler equations gives rise
to a fundamental solution with initial data imposed to the line $\rho=1$. This
is in contrast to the common approach (when $\gamma >1$) which prescribes
initial data on the vacuum line $\rho =0$. The entropies we construct here are
weak entropies, i.e. they vanish when the density vanishes. Another feature of
our proof lies in the reduction theorem which makes use of the family of weak
entropies to show that a Young measure must reduce to a Dirac mass. This step
is based on new convergence results for regularized products of measures and
functions of bounded variation.
