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

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

2012-01-29

arxiv.org/abs/1201.6009v2

Mathematics

Analysis of PDEs

A revised version with improved notation, proofs, etc. 19 pages

Scientific paper

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.

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