IUMJ

Title: Sonic-supersonic solutions for the steady Euler equations

Authors: Tianyou Zhang and Yuxi Zheng

Issue: Volume 63 (2014), Issue 6, 1785-1817

Abstract:

Given a smooth curve as a sonic line in the plane, we construct a local smooth supersonic solution on one side of the curve for the steady compressible Euler system of equations in two space dimensions. Our construction hinges on a new set of coordinates introduced here to handle the inherent degeneracy of the system at the sonic curve. We analyze the streamlines of the solutions to illustrate that the shock-free portion of the solutions may be combined with known results of existence of sonic-subsonic solutions of Xie and Xin [Chunjing Xie and Zhouping Xin, \textit{Global subsonic and subsonic-sonic flows through infinitely long nozzles}, Indiana Univ. Math. J. \textbf{56} (2007), no. 6, 2991--3023] on the other side of the curve to form shock-free transonic flows in a channel. The existence result is also a partial generalization of the exact solution of Ringleb [F. Ringleb, \textit{Exacte L\"osungen der Differentialgleichunsen eineradiabatischen Gasstr\"omung}, ZAMM \textbf{20} (1940), 185--198] toward a flexible existence.