Conditions Implying Energy Equality for Weak Solutions of the Navier--Stokes Equations

TL;DR

This paper introduces new conditions on weak solutions of the Navier--Stokes equations that ensure energy equality, especially when the singularity set has low dimension, extending classical results and including fractional Laplacian cases.

Contribution

It presents novel $L^q L^p$ conditions guaranteeing energy equality for weak solutions with low-dimensional singularities, surpassing classical criteria and covering fractional Laplacian cases.

Findings

New $L^q L^p$ conditions ensure energy equality for solutions with singularity set $d<1$.

Energy equality established for certain Type-I blowup solutions.

Results extended to Navier--Stokes equations with fractional Laplacian below order 1.

Abstract

When a Leray--Hopf weak solution to the NSE has a singularity set $S$ of dimension $d$ less than $3$---for example, a suitable weak solution---we find a family of new $L^q L^p$ conditions that guarantee validity of the energy equality. Our conditions surpass the classical Lions--Lady\v{z}enskaja $L^4 L^4$ result in the case $d<1$. Additionally, we establish energy equality in certain cases of Type-I blowup. The results are also extended to the NSE with fractional power of the Laplacian below $1$.