On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

TL;DR

This paper proves the global existence and regularity of solutions to the 2-D steady Euler equations in supersonic-sonic patches, analyzing behavior near sonic curves with new mathematical techniques.

Contribution

It introduces new characteristic decompositions and a local partial hodograph transformation to establish solution existence and regularity near sonic curves.

Findings

Solutions are uniformly $C^{1,1/6}$ continuous up to sonic curves.

Sonic curves are shown to be $C^{1,1/6}$ continuous.

Established global solutions for the full 2-D steady Euler system in supersonic-sonic patches.

Abstract

Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,\frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,\frac{1}{6}}$ continuous.