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We study the Cauchy problem associated with the system of two conservationlaws arising in isothermal gas dynamics, in which the pressure and the densityare related by the $\gamma$-law equation $p(\rho) \sim \rho^\gamma$ with$\gamma =1$. Our results complete those obtained earlier for $\gamma >1$. Weprove the global existence and compactness of entropy solutions generated bythe vanishing viscosity method. The proof relies on compensated compactnessarguments and symmetry group analysis. Interestingly, we make use here of thefact that the isothermal gas dynamics system is invariant modulo a linearscaling of the density. This property enables us to reduce our problem to thatwith a small initial density. One symmetry group associated with the linearhyperbolic equations describing all entropies of the Euler equations gives riseto a fundamental solution with initial data imposed to the line $\rho=1$. Thisis in contrast to the common approach (when $\gamma >1$) which prescribesinitial data on the vacuum line $\rho =0$. The entropies we construct here areweak entropies, i.e. they vanish when the density vanishes. Another feature ofour proof lies in the reduction theorem which makes use of the family of weakentropies to show that a Young measure must reduce to a Dirac mass. This stepis based on new convergence results for regularized products of measures andfunctions of bounded variation.Comment: 29 page

Symmetries and Global Solvability of the Isothermal Gas Dynamics Equations