A study of energy concentration and drain in incompressible fluids

In this paper, we examine two opposite scenarios of energy behaviour for solutions of the Euler equation. We show that if u is a regular solution on a time interval [0, T) and if u ∈ LrL∞ for some r≥ 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 smaller than N - (2)/r-1. The same holds for solutions of the three-dimensional Navier-Stokes equation in the range 5/3 < r < 7/4. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than (T − t)1−δ, for any δ > 0. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature. (paper)