The ellipticity principle for steady and selfsimilar polytropic potential flow∗

We prove the ellipticity principle for selfsimilar potential flows for gas dynamics. We show that the interior of a pseudo-subsonic-sonic-region of a smooth solution must be pseudo-subsonic. In fact, the pseudo-Mach number is below that of a domain-dependent function which is < 1 in the interior and ≤ 1 on the boundary. Therefore the interior must stay pseudo-subsonic under homotopy of pseudo-subsonic-sonic boundaries. Self-similar flows represent time-asymptotic flow patterns for flows with self-similar solid boundary and far field values. Our result indicates that such a flow do not have pseudo-supersonic bubbles either away from boundary or along a straight solid boundary within a pseudo-subsonic region. This is in contrast with stationary flows, for which a supersonic bubble can arise within a subsonic region and eventually forms shocks. Our analysis is for polytropic gases. We give two examples showing that the ellipticity principle does not extend to the isentropic and full Euler equations, or to arbitrary pressure laws.