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

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.

arXiv commentA revised version with improved notation, proofs, etc. 19 pages

References in corpus (2)

• Nonexistence of self-similar singularities for the 3D incompressible Euler equations
• A note on `Nonexistence of self-similar singularities for the 3D incompressible Euler equations'

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• Continuation of the zero set for discretely self-similar solutions to the Euler equations
• 2D homogeneous solutions to the Euler equation
• Remarks on the asymptotically discretely self-similar solutions of the Navier-Stokes and the Euler equations
• Energy concentrations and Type I blow-up for the 3D Euler equations