A study of energy concentration and drain in incompressible fluids

TL;DR

This paper investigates energy concentration and dissipation in solutions of the Euler and Navier-Stokes equations, establishing conditions that prevent energy concentration and characterize energy decay near singularities.

Contribution

It provides new bounds on energy concentration sets for regular solutions and characterizes energy decay rates, extending understanding of singularity formation in fluid dynamics.

Findings

Energy cannot concentrate on small Hausdorff dimension sets under certain regularity conditions.

Energy must decay faster than a specific rate if it vanishes in a subregion.

New exclusions of self-similar blow-up scenarios are derived.

Abstract

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 $5/3<r<7/4$. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than $(T-t)^{1-\d}$, for any $\d>0$. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.