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arXiv · 1606.02363

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

Trevor M. Leslie·Roman Shvydkoy
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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 $1$.

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Trevor M. Leslie, Roman Shvydkoy. 2016-06-08. Conditions Implying Energy Equality for Weak Solutions of the Navier--Stokes Equations. https://arxiv.org/abs/1606.02363

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