The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

Yanbo Hu; Tong Li

doi:10.3934/cpaa.2019149

Article Contents

2019, Volume 18, Issue 6: 3317-3336. Doi: 10.3934/cpaa.2019149

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The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

Yanbo Hu^1, , and
Tong Li^2,

1.
Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China
2.
Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

^* Corresponding author
^* Corresponding author

Received: July 31, 2018

Revised: January 31, 2019

Published: May 2019

The first author was supported by NSF of Zhejiang Province LY17A010019, NSFC 11301128, 11571088 and China Scholarship Council 201708330155

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Abstract

We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly ${C^{1,\frac{1}{6}}}$ up to the degenerate sonic boundary and that the sonic curve is ${C^{1,\frac{1}{6}}}$.

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Mathematics Subject Classification: 35L65, 35L80, 35R35.

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Figure 1. The semi-hyperbolic patch

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Figure 2. Case 2

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Figure 3. The region of $ \Omega_\nu(\bar{z}) $

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