1205.1544 (Roman Shvydkoy)

A study of energy concentration and drain in incompressible fluids    [PDF]

Roman Shvydkoy

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 $u \in L^rL^\infty$ 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 $5/30$. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.

View original: http://arxiv.org/abs/1205.1544