Self-similar solutions are considered to the incompressible Euler equations in R 3, where the similarity variable is defined as ~ = x/(T -t) f~ E R a, ~ _ 0. It is shown that the scaling exponent is bounded above: 3 _< 1. Requiring [[ui[ยฃu < oa and allowing more than one length scale, it is found/3 E [2/5, 1]. This new result on the self-similar singularity is consistent with known analytical results for blow-up conditions. (~) 2000 Elsevier Science Ltd. All rights reserved.
Whether smooth solutions of the incompressible Euler equations in three space dimensions develop finite-time singularities remains an open question. Some numerical studies and asymptotic models support the possibility (see for example, an earlier paper [1] and [2] with references therein). Rigorous analyses in [3,4] give necessary and sufficient conditions for singularity formation.
A natural assumption to make about the singularity is its self-similar form. This was first advanced by Leray [5] for the 3D incompressible Navier-Stokes equations. But for the Leray system, no such bounded solutions were found [6], while [7] proves that the only solution in/:3(R3) is trivial. However, the result of [7] does not exclude self-similar solutions which locally satisfy the natural energy estimates. There has been continued interest in searching for self-similar Navier-Stokes singularities [8].
In this letter, we consider self-similar transformations of the 3D Euler equation, where the transformation variable is defined by ~ = x/(T -t) 3, 3 >_ O. We show that for existence of a self-similar Euler singularity at time t = T with finite energy, there are necessary bounds on the scaling exponent: /3 E [2/5, 1]. Recent works [9-13] indicate self-similar scalings for Euler singularities, it is then expected that our results here may be complementary with numerical solutions for further investigation.