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

DOI10.1007/S00205-013-0630-ZzbMATH Open1285.35070arXiv1201.6009OpenAlexW2072138519MaRDI QIDQ394016

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Publication date: 24 January 2014

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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 $uin{L}^p$ and $usimfrac1t^{a/(1+a)}$, then the blow-up does not occur provided $a>N/2$ or $-1<aleqN/p$. 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.

Full work available at URL: https://arxiv.org/abs/1201.6009

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