Electron. J. Diff. Equ., Vol. 2015 (2015), No. 243, pp. 1-20.

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

Ilija Jegdic, Katarina Jegdic

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.

Submitted April 27, 2015. Published September 22, 2015.
Math Subject Classifications: 35L65.
Key Words: Unsteady transonic small disturbance equation; mixed type system; semi-hyperbolic patch; Goursat-type problem.

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Ilija Jegdic
Department of Mathematics and Physics
Houston Baptist University
7502 Fondren Rd, Houston, TX 77074, USA
email: i_jegdic@yahoo.com
Katarina Jegdic
Department of Mathematics and Statistics
University of Houston - Downtown
1 Main St, Houston, TX 77002, USA
email: jegdick@uhd.edu

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