January & February  2009, 23(1&2): 605-616. doi: 10.3934/dcds.2009.23.605

Absorption of characteristics by sonic curve of the two-dimensional Euler equations

1. 

Department of Mathematics, The Pennsylvania State University, PA 16802, United States

Received  January 2008 Revised  April 2008 Published  September 2008

We explore the reflection off a sonic curve and the domain of determinacy, via the method of characteristics, of self-similar solutions to the two dimensional isentropic Euler system through several examples with axially symmetric initial data. We find that characteristics in some cases can be completely absorbed by the sonic curve so that the characteristics vanish tangentially into the sonic boundary, exemplifying a classical scenario of the Keldysh type; however, the characteristics can wrap around the closed sonic curve unboundedly many times, so that the domain of determinacy of the hyperbolic characteristic boundary value problem or the Goursat problem exhibit layered annulus structures. As the number of layers increases, the layers become thinner, and the solution at an interior point of the domain depends eventually on the entire boundary data.
Citation: Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605
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