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Potential Singularity for a Family of Models of the Axisymmetric Incompressible Flow

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Abstract

We study a family of 3D models for the incompressible axisymmetric Euler and Navier–Stokes equations. The models are derived by changing the strength of the convection terms in the equations written using a set of transformed variables. The models share several regularity results with the Euler and Navier–Stokes equations, including an energy identity, the conservation of a modified circulation quantity, the BKM criterion and the Prodi–Serrin criterion. The inviscid models with weak convection are numerically observed to develop stable self-similar singularity with the singular region traveling along the symmetric axis, and such singularity scenario does not seem to persist for strong convection.

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Acknowledgements

This research was in part supported by NSF Grants DMS-1613861 and DMS-1318377. The research of T. Jin was supported in part by Hong Kong RGC grant ECS 26300716. Part of this work was done when T. Jin was visiting California Institute of Technology as an Orr foundation Caltech-HKUST Visiting Scholar. He would like to thank Professor Thomas Y. Hou for hosting his visit.

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Authors and Affiliations

  1. Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Thomas Y. Hou & Pengfei Liu

  2. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Tianling Jin

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  1. Thomas Y. Hou
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  2. Tianling Jin
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  3. Pengfei Liu
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Correspondence to Pengfei Liu.

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Communicated by Alex Kiselev.

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Hou, T.Y., Jin, T. & Liu, P. Potential Singularity for a Family of Models of the Axisymmetric Incompressible Flow. J Nonlinear Sci 28, 2217–2247 (2018). https://doi.org/10.1007/s00332-017-9370-9

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  • DOI: https://doi.org/10.1007/s00332-017-9370-9

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