1-D Isentropic Euler flows: Self-similar Vacuum Solutions

We consider one-dimensional self-similar solutions to the isentropic Euler system when the initial data are at vacuum to the left of the origin. For $x>0$ the initial velocity and sound speed are of form $u_0(x)=u_+x^{1-\lambda}$ and $c_0(x)=c_+x^{1-\lambda}$, for constants $u_+\in\RR$, $c_+>0$, $\lambda\in\RR$. We analyze the resulting solutions in terms of the similarity parameter $\lambda$, the adiabatic exponent $\gamma$, and the initial (signed) Mach number $\text{Ma}=u_+/c_+$. Restricting attention to locally bounded data, we find that when the sound speed initially decays to zero in a H\"older manner ($0<\lambda<1$), the resulting flow is always defined globally. Furthermore, there are three regimes depending on $\text{Ma}$: for sufficiently large positive $\text{Ma}$-values, the solution is continuous and the initial H\"older decay is immediately replaced by $C^1$-decay to vacuum along a stationary vacuum interface; for moderate values of $\text{Ma}$, the solution is again continuous and with an accelerating vacuum interface along which $c^2$ decays linearly to zero (i.e., a "physical singularity''); for sufficiently large negative $\text{Ma}$-values, the solution contains a shock wave emanating from the initial vacuum interface and propagating into the fluid, together with a physical singularity along an accelerating vacuum interface. In contrast, when the sound speed initially decays to zero in a $C^1$ manner ($\lambda<0$), a global flow exists only for sufficiently large positive values of $\text{Ma}$. Non-existence of global solutions for smaller $\text{Ma}$-values is due to rapid growth of the data at infinity and is unrelated to the presence of a vacuum.