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November  2019, 18(6): 3317-3336. doi: 10.3934/cpaa.2019149

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

 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

Received  August 2018 Revised  February 2019 Published  May 2019

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

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 ${C^{1,\frac{1}{6}}}$.

Citation: Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149
##### References:

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##### References:
The semi-hyperbolic patch
Case 2
The region of $\Omega_\nu(\bar{z})$
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