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 u₀(x)=u₊x^1-λ and c₀(x)=c₊x^1-λ, for constants u₊∈R, c₊>0, λ∈R. We analyze the resulting solutions in terms of the similarity parameter λ, the adiabatic exponent γ, and the initial (signed) Mach number 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<λ<1), the resulting flow is always defined globally. Furthermore, there are three regimes depending on Ma: for sufficiently large positive Ma-values, the solution is continuous and the initial Hölder decay is immediately replaced by C¹-decay to vacuum along a stationary vacuum interface; for moderate values of Ma, the solution is again continuous and with an accelerating vacuum interface along which c² decays linearly to zero (i.e., a “physical singularity”); for sufficiently large negative 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 C¹ manner (λ<0), a global flow exists only for sufficiently large positive values of Ma. The non-existence of global solutions for smaller Ma-values is due to rapid growth of the data at infinity and is unrelated to the presence of a vacuum.