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

摘要

It is shown that the self-similar ansatz proposed by T. Hou and G. Luo todescribe a singular solution of the 3D axisymmetric Euler equations leads,without assuming any asymptotic condition on the self-similar profiles, to anover-determined system of partial differential equations that produces twofamilies of solutions: a class of trivial solutions in which the vorticityfield 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, theanalytical properties of these solutions are not consistent with the numericalobservations reported by T. Hou and G. Luo. Therefore, this result is arefinement of the previous work published by D. Chae and T.-P. Tsai on thismatter, where the authors found the trivial class of solutions under a certaindecay condition of the blow-up profiles.