Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions

Eun Heui Kim; Charis Tsikkou

doi:10.3934/dcds.2017271

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2017, Volume 37, Issue 12: 6257-6289. Doi: 10.3934/dcds.2017271

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Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions

Eun Heui Kim^1, , and
Charis Tsikkou^2,

1.
Department of Mathematics and Statistics, California State University, Long Beach, Long Beach, CA 90840, USA
2.
Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA

* Corresponding author: Eun Heui Kim.
* Corresponding author: Eun Heui Kim.

Received: December 31, 2016

Revised: May 31, 2017

Published: October 2017

The work of Kim was supported by the National Science Foundation under the Grants DMS- 1109202, 1615266. The work of Tsikkou was supported by the National Science Foundation under the Grant DMS-1400168.

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Abstract

We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.

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Mathematics Subject Classification: Primary: 76L05, 35L65; Secondary: 65M06, 35M33.

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Figure 1. Riemann data and configuration.

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Figure 2. Configuration with details.

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Figure 3. Regions and characteristics in the supersonic region.

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Figure 4. $\mathcal{R}_1\setminus \mathcal{R}_1[\delta]$, the region $\mathcal{R}_1$ excluding the small neighborhoods of $\Xi_1$ and $\Xi_3.$

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Figure 5. Schematics of constructing the solution near $\Xi_3.$

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Figure 6. Envelope formation by the simple wave.

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Figure 7. Density plots: the contour plot of $\rho$.

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Figure 8. Density plots: Left figure is the cross section in the radial direction for a fixed angle $\theta$ ranging from $0$ to $\pi/2$ and incrementing by $10$ degrees. Right figure is the enlargement of the left figure near which the shock changes to sonic, which appears to be in-between the angle $50$ and $60$ degrees in this configuration.

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