On formation of a locally self-similar collapse in the incompressible Euler equations

TL;DR

This paper investigates the possibility of locally self-similar blow-up solutions in the incompressible Euler equations, providing conditions under which such blow-ups cannot occur based on integrability and scaling properties.

Contribution

It establishes new exclusion results for self-similar blow-up solutions in the Euler equations under various $L^p$ and scaling conditions, extending understanding of singularity formation.

Findings

Self-similar blow-up does not occur if $u otin L^p$ with certain scaling exponents.

Profiles with specific asymptotic power bounds are excluded.

Homogeneous solutions at infinity are eliminated unless scaling invariant.

Abstract

The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling exponents. In particular, in $N$ dimensions if in self-similar variables $u \in L^p$ and $u \sim \frac{1}{t^{\a/(1+\a)}}$, then the blow-up does not occur provided $\a >N/2$ or $-1<\a\leq N/p$. This includes the $L^3$ case natural for the Navier-Stokes equations. For $\a = N/2$ we exclude profiles with an asymptotic power bounds of the form $ |y|^{-N-1+\d} \lesssim |u(y)| \lesssim |y|^{1-\d}$. Homogeneous near infinity solutions are eliminated as well except when homogeneity is scaling invariant.