IUMJ

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

Authors: Anne Bronzi and Roman Shvydkoy

Issue: Volume 64 (2015), Issue 5, 1291-1302

Abstract:

In this note, we study locally self-similar blowup for the Euler equation. The main result states that under a mild $L^p$-growth assumption on the profile $v$, namely $\displaystyle\int_{|y|\sim L}|v|^p\,\mathrm{d}y\lesssim L^{\gamma}$ for some $\gamma<p-2$, the self-similar solution carries a positive amount of energy up to the time of blowup $T$, namely, $\displaystyle\int_{|y|\sim L}|v|^2\,\mathrm{d}y\sim L^{N-2\alpha}$. 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.