Admissible states and pressure for one-dimensional ideal-gas Euler

Fix $\gamma >1$. In one space dimension the conserved state is $U=(\rho ,m,E)\in {\mathbb{R}}^3$, where $\rho$ is density, $m$ momentum, and $E$ total energy per unit volume. The admissible-state set is the open set ${\mathcal{G}}_{\gamma }=\{(\rho ,m,E):\rho >0,E>m^2\mathrm{/}(2\rho )\}\mathrm{.}$ The ideal-gas pressure on this set is $p(U)=(\gamma -1)(E-\frac{m^2}{2\rho })>0,$ the velocity is $u=m\mathrm{/}\rho$, and the flux is $F(U)=(m,\frac{m^2}{\rho }+p,\frac{m}{\rho }(E+p))\mathrm{.}$ The sound speed is $c=\sqrt{\gamma p\mathrm{/}\rho }$. These definitions apply only on ${\mathcal{G}}_{\gamma }$; a state with nonpositive density or nonpositive internal energy is inadmissible. The set is open because $\rho$ and $E-m^2\mathrm{/}(2\rho )$ are continuous on $\rho >0$ and the inequalities are strict; all displayed flux functions are smooth on this domain.