A study of energy concentration and drain in incompressible fluids

In this paper we examine two opposite scenarios of energy behavior for solutions of the Euler equation. We show that if $u$ is a regular solution on a time interval $[0,T)$ and if $u\in L^r{L}^{\mathrm{\infty }}$ for some $r\geq \frac{2}{N}+1$,where$N$ is the dimension of the fluid, then the energy at the time $T$ cannot concentrate on a set of Hausdorff dimension samller than $N - \frac{2}{r-1}$. The same holds for solutions of the three-dimensional Navier-Stokes equation in the range $5\mathrm{/}3<r<7\mathrm{/}4$. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than $(T-t{)}^{1-\text{d}}$, for any $\text{d}>0$. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.