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Two-Dimensional Pseudosteady Flows Around a Sharp Corner

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Abstract

We consider two-dimensional (2D) pseudosteady flows around a sharp corner. This problem can be seen as a 2D Riemann initial and boundary value problem (IBVP) for the compressible Euler system. The initial state is a combination of a uniform flow in one quadrant and vacuum in the remaining domain. The boundary condition on the wall of the sharp corner is a slip boundary condition. By a self-similar transformation, the 2D Riemann IBVP is converted into a boundary value problem (BVP) for the 2D self-similar Euler system. Existence of global piecewise smooth (or Lipshitz-continuous) solutions to the BVP are obtained. One of the main difficulties for the global existence is that the type of the 2D self-similar Euler system is a priori unknown. In order to use the method of characteristic analysis, we establish some a priori estimates for the hyperboliciy of the system. The other main difficulty is that when the uniform flow is sonic or subsonic, the hyperbolic system becomes degenerate at the origin. Moreover, there is a multi-valued singularity at the origin. To solve this degenerate hyperbolic boundary value problem, we establish some uniform interior \(C^{0, 1}\) norm estimates for the solutions of a sequence of regularized hyperbolic boundary value problems, and then use the Arzela–Ascoli theorem and a standard diagonal procedure to construct a global Lipschitz continuous solution. The method used here may also be used to construct continuous solutions of some other degenerate hyperbolic boundary value problems and sonic-supersonic flow problems.

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Acknowledgements

Both authors would like to thank the anonymous referees for their careful reading on the original manuscript and helpful suggestions and comments, which greatly improve the presentation of the paper.

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Geng Lai & Wancheng Sheng

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  1. Geng Lai
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  2. Wancheng Sheng
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Correspondence to Wancheng Sheng.

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Communicated by T. Liu.

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The research of the first author was supported by NSFC 12071278. The research of the second author was supported by NSFC 11771274.

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Lai, G., Sheng, W. Two-Dimensional Pseudosteady Flows Around a Sharp Corner. Arch Rational Mech Anal 241, 805–884 (2021). https://doi.org/10.1007/s00205-021-01665-0

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