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arXiv · 1310.8611

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

Anne Bronzi·Roman Shvydkoy
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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, $\int_{|y| \sim L} |v|^p dy \lesssim L^{\g}$ for some $\g <p-2$, the self-similar solution carries a positive amount of energy up to the time of blow-up $T$, namely, $\int_{|y| \sim L} |v|^2 dy \sim L^{N-2\a}$. 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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Anne Bronzi, Roman Shvydkoy. 2014-09-15. On the energy behavior of locally self-similar blowup for the Euler equation. https://arxiv.org/abs/1310.8611

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