Well-posedness of non-isentropic Euler equations with physical vacuum

We consider the local well-posedness of the one-dimensional nonisentropic Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation laws becomes characteristic and degenerate at the vacuum boundary. Our proof is based on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term. Then we construct the solutions to this degenerate parabolic problem and establish the estimates that are uniform with respect to the artificial viscosity parameter. Solutions to the compressible Euler equations are obtained as the limit of the vanishing artificial viscosity. Different from the isentropic case \cite{Coutand4, Lei}, our momentum equation of conservation laws has an extra term $p_{S}S_x$ that leads to some extra terms in the energy function and causes more difficulties even for the case of $\gamma=2$. Moreover, we deal with this free boundary problem starting from the general cases of $2\leq\gamma<3$ and $1<\gamma<2 $ instead of only emphasizing the isentropic case of $\gamma=2$ in \cite{Coutand4, jang1, Lei}.