On the energy behavior of locally self-similar blowup for the Euler equation

2013/10/31 by Anne Bronzi, Bronzi, Anne, Roman Shvydkoy +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1310.8611

an update of the previous version

arxiv created 2014/09/15 · arxiv updated 2014/09/16

Abstract

In this note we study locally self-similar blow up for the Euler equation. The main result states that under a mild L^p-growth assumption on the profile v, namely, ∫_{|y| ∼ L} |v|^p dy \lesssim L^\g for some \g <p-2, the self-similar solution carries a positive amount of energy up to the time of blow-up T, namely, ∫_{|y| ∼ L} |v|² dy ∼ L^N-2\a. The result implies and extends several previously known exclusion criteria. It also supports a general conjecture relating fractal local dimensions of the energy measure with the rate of velocity growth at the time of possible blowup.

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