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

Thomas Yizhao Hou () and Pengfei Liu ()
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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

Chapter 17 in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 2018, pp 869-899 from Springer

Abstract: 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.

Keywords: Self-similar Profile; Finite Time Singularity; Singular Solutions; Dynamic Rescaling; Periodic Cylinder (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-13344-7_17

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DOI: 10.1007/978-3-319-13344-7_17

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