article

Title: The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics

Authors: Qin Wang and Yuxi Zheng

Issue: Volume 63 (2014), Issue 2, 385-402

Abstract:

$We study the uniform regularity of semi-hyperbolic patches of self-similar solutions near sonic lines to a general Riemann problem for the pressure gradient equation. This type of solution, in which one family of characteristics starts on a sonic line and ends on a transonic shock wave, is common for the Riemann problems for the Euler system in two space dimensions. The global existence of smooth solutions was established in Song and Zheng [Disc.\ Cont.\ Dyna.\ Syst., \textbf{24} (2009),1365-1380], but the smoothness near the sonic lines is not clear. We establish that the smooth solutions are uniformly smooth up to their sonic boundaries, and that the sonic lines are $C^1$ continuous.$