Symmetries and global solvability of the isothermal gas dynamics equations

Abstract

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 $γ$-law equation $p(ρ) \sim ρ^γ$ with $γ=1$. Our results complete those obtained earlier for $γ>1$. 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 $ρ=1$. This is in contrast to the common approach (when $γ>1$) which prescribes initial data on the vacuum line $ρ=0$. 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.