The ellipticity principle for the 3-D steady potential flow in conical coordinates

Abstract

In this paper, we consider the 3-D steady potential flow for a compressible gas with pressure satisfying $p&apos;(\\rho)=\\rho^{\\gamma-1}$, where $\\rho$ is the density and $\\gamma\\geq-1$ is a constant. The potential equation is of mixed type in conical coordinates, and the type is completely determined by a pseudo-Mach number $L$ with $L<1$ (resp. $L>1$) corresponding to elliptic (resp. hyperbolic) regions. We first establish an ellipticity principle: for $\\gamma\\geq-1$, the equation is elliptic in the interior of a parabolic-elliptic region; in particular, when $\\gamma>-1$, there exists a domain-dependent function $\\delta_0>0$ such that $L\\leq~1-\\delta_0$. Then applying this ellipticity principle, we further prove that for $\\gamma\\geq-1$, the equation is uniformly elliptic in any subregion that is strictly away from the degenerate boundary.