Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Abstract

In this paper we rule out the possibility of asymptotically selfsimilar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T . For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm,Ḃ 0 1,∞ (R 3 ). For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in L q (B(z, r)) for some q ∈ [2, ∞), where the ball of radius r is shrinking toward a possible singularity point z at the order of √ T − t as t approaches to T . In the L q (R 3 ) convergence case with q ∈ [3, ∞) we present a simple alternative proof of the similar result in [16]. *

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

“…], see also [20] for 'pseudo self-similar solutions'). A study of self-similar blow-up in the settings adopted here was undertaken by the first author in a series of works [6,7,5,4]. The main two results obtained are the following…”

Self-similar singularity of a 1D model for the 3D axisymmetric Euler equations

“…As a result, the velocity field at the singularity time is Hölder continuous. Such behavior was also observed in the numerical simulation of the 3D Euler equations in [15], which is very different from the Leray type of self-similar solutions of the 3D Euler equations, whose existence has been ruled out under certain decay assumptions on the self-similar profiles [3][4][5].…”

Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations

“…We remark that Dr. Chae, motivated by the result presented in this paper, has recently obtained more general nonexistence results for asymptotically self-similar singularities in the Euler and Navier-Stokes equations [3]. For more discussions regarding other aspects of the Navier-Stokes equations, we refer the reader to [5,20,16].…”