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 the form \(u_0(x)=u_+x^{1-\lambda }\) and \(c_0(x)=c_+x^{1-\lambda }\), for constants \(u_+\in \mathbb {R}\), \(c_+>0\), \(\lambda \in \mathbb {R}\). 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 Hölder 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 Hölder 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 \(\text {Ma}\). The 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.