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The Free Boundary of Steady Axisymmetric Inviscid Flow with Vorticity I: Near the Degenerate Point

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Abstract

In this paper, we investigate the singularity near the degenerate points of the steady axisymmetric flow with general vorticity of an inviscid incompressible fluid acted on by gravity and with a free surface. We called the points on the free boundary at which the gradient of the stream function vanishes as the degenerate points. The main results in this paper give the different classifications of the singularity near the degenerate points on the free surface. More precisely, we obtained that at the stagnation points, the possible profiles must be a Stokes corner, a horizontal cusp, or a horizontal flatness. At the degenerate points on the symmetric axis except the origin, the wave profile must be a cusp. At the origin, the possible wave profiles must be a Garabedian pointed bubble, a horizontal cusp, or a horizontal flatness.

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Acknowledgements

The authors would like to thank the anonymous referees for careful reading and for the valuable comments.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, People’s Republic of China

    Lili Du

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Lili Du & Jinli Huang

  3. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Yang Pu

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  1. Lili Du
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  2. Jinli Huang
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Correspondence to Yang Pu.

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Du and Huang are supported by National Nature Science Foundation of China Grant 11971331, 12125102, and Sichuan Youth Science and Technology Foundation 2021JDTD0024. Pu is supported by National Nature Science Foundation of China Grant 12201504.

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Du, L., Huang, J. & Pu, Y. The Free Boundary of Steady Axisymmetric Inviscid Flow with Vorticity I: Near the Degenerate Point. Commun. Math. Phys. 400, 2137–2179 (2023). https://doi.org/10.1007/s00220-023-04651-7

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  • DOI: https://doi.org/10.1007/s00220-023-04651-7

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