# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021015

## On a supersonic-sonic patch arising from the frankl problem in transonic flows

 1 Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China 2 Laboratory of Computational Physics, Institute of Applied Physics, and Computational Mathematics, Beijing, 100088, China 3 Center for Applied Physics and Technology, Peking University, 100871, China

* Corresponding author

Dedicated to the celebration of the 80th birthday of Professor Shuxing Chen

Received  September 2020 Revised  December 2020 Published  February 2021

Fund Project: The first author is supported by the Zhejiang Provincial Natural Science Foundation (No. LY21A010017). The second author is supported by the Natural Science Foundation of China (Nos: 11771054, 91852207, 12072042), National Key Project(GJXM92579) and Foundation of LCP

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity of solutions in a partial hodograph coordinate system in terms of angle variables. The original problem is solved by transforming the solution in the partial hodograph plane back to that in the physical plane. Moreover, the uniform regularity of the solution and the regularity of an associated sonic curve are also verified.

Citation: Yanbo Hu, Jiequan Li. On a supersonic-sonic patch arising from the frankl problem in transonic flows. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021015
##### References:

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##### References:
Transonic phenomena in a duct
The modified Frankl problem. With a velocity distribution on the arcs $\widehat{PE}$ and $\widehat{FQ}$, find an airfoil's arc $\widehat{EF}$, free of boundary conditions, for the correctness of the problem in the class of smooth solutions
The region $\Omega_ \varepsilon$
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