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Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations

In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Author

Listed:
  • Thomas Yizhao Hou

    (California Institute of Technology, Division of Engineering and Applied Science, Computing and Mathematical Sciences Department)

  • Pengfei Liu

    (California Institute of Technology, Computing and Mathematical Sciences)

Abstract

Global regularity of the Euler equations in the three-dimensional (3D) setting is regarded as one of the most important open questions in mathematical fluid mechanics. In this work we consider two one-dimensional (1D) models approximating the dynamics of the 3D axisymmetric Euler equations on the solid boundary of a periodic cylinder, which are motivated by a potential finite-time singularity formation scenario proposed recently by Luo and Hou (PNAS 111(36):12968–12973, 2014), and numerically investigate the stability of the self-similar profiles in their singular solutions. We first review some recent existence results about the self-similar profiles for one model, and then derive the evolution equations of the spatial profiles in the singular solutions for both models through a dynamic rescaling formulation. We demonstrate the stability of the self-similar profiles by analyzing their discretized dynamics using linearization, and it is hoped that these computations can help to understand the potential singularity formation mechanism of the 3D Euler equations.

Suggested Citation

  • Thomas Yizhao Hou & Pengfei Liu, 2018. "Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations," Springer Books, in: Yoshikazu Giga & Antonín Novotný (ed.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, chapter 17, pages 869-899, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-13344-7_17
    DOI: 10.1007/978-3-319-13344-7_17
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