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arXiv · 1708.09648

Further remarks on the Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations

Gianmarco Sperone
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Abstract

It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a singular solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profiles, to an over-determined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consistent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refinement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors found the trivial class of solutions under a certain decay condition of the blow-up profiles.

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Gianmarco Sperone. 2017-08-31. Further remarks on the Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations. https://doi.org/10.1007/s00332-017-9363-8

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