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(+ )is an element of R, c(+) > 0, lambda is an element of R. We analyze the resulting solutions in terms of the similarity parameter lambda, the adiabatic exponent gamma, and the initial (signed) Mach number M-a = u(+)/c(+). Restricting attention to locally bounded data, we find that when the sound speed initially decays to zero in a H & ouml;lder manner (0 < lambda < 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 H & ouml;lder decay is immediately replaced by C-1-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 c(2) 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 C-1 manner (lambda < 0), a global flow exists only for sufficiently large positive values of Ma. 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.