Properties of solutions in semi-hyperbolic patches for unsteady transonic small disturbance equations

• Main Authors: Ilija Jegdic, Katarina Jegdic
• Format: Article
• Language: English
• Published: Texas State University 2015-09-01
• Series: Electronic Journal of Differential Equations
• Subjects: Unsteady transonic small disturbance equation; mixed type system; semi-hyperbolic patch; Goursat-type problem
• Online Access: http://ejde.math.txstate.edu/Volumes/2015/243/abstr.html

We consider a two-dimensional Riemann problem for the unsteady transonic small disturbance equation resulting in diverging rarefaction waves. We write the problem in self-similar coordinates and we obtain a mixed type (hyperbolic-elliptic) system. Resolving the one-dimensional discontinuities in the far field, where the system is hyperbolic, and using characteristics, we formulate the problem in a semi-hyperbolic patch that is between the hyperbolic and the elliptic regions. A semi-hyperbolic patch is known as a region where one family out of two nonlinear families of characteristics starts on a sonic curve and ends on a transonic shock. We obtain existence of a smooth local solution in this semi-hyperbolic patch and we prove various properties of global smooth solutions based on a characteristic decomposition using directional derivatives.