The ellipticity principle for selfsimilar polytropic potential flow

TL;DR

This paper proves an ellipticity principle for self-similar potential flow of compressible gases, showing that parabolic-elliptic regions cannot be open and providing bounds on the pseudo-Mach number within solutions.

Contribution

It establishes a rigorous ellipticity principle for self-similar potential flow, ruling out open parabolic regions and characterizing the elliptic regions in such flows.

Findings

Interior of parabolic-elliptic regions must be elliptic

Pseudo-Mach number L is bounded away from 1 in these regions

No open parabolic regions can exist in smooth solutions

Abstract

We consider self-similar potential flow for compressible gas with polytropic pressure law. Self-similar solutions arise as large-time asymptotes of general solutions, and as exact solutions of many important special cases like Mach reflection, multidimensional Riemann problems, or flow around corners. Self-similar potential flow is a quasilinear second-order PDE of mixed type which is hyperbolic at infinity (if the velocity is globally bounded). The type in each point is determined by the local pseudo-Mach number L, with L<1 resp. L>1 corresponding to elliptic resp. hyperbolic regions. We prove an ellipticity principle: the interior of a parabolic-elliptic region of a sufficiently smooth solution must be elliptic; in fact $L$ must be bounded above away from 1 by a domain-dependent function. In particular there are no open parabolic regions. We also discuss the case of slip boundary conditions at straight solid walls.