On the energy and helicity conservation of the incompressible Euler equations

TL;DR

This paper establishes minimal regularity conditions for weak solutions of the incompressible Euler equations to conserve energy and helicity, extending previous results and highlighting the importance of integrability and regularity.

Contribution

It generalizes classical conservation results by identifying new minimal regularity conditions in Besov spaces for energy and helicity conservation.

Findings

Energy is conserved if velocity in L^p(0,T;B^{1/p}_{2p/(p-1),c(N)}) with 1<p≤3.

Helicity is conserved if velocity in L^p(0,T;B^{2/p}_{2p/(p-1),c(N)}) with 2<p≤3.

Results hold for both periodic domain and whole space.

Abstract

In this paper, we are concerned with the minimal regularity of weak solutions implying the law of balance for both energy and helicity in the incompressible Euler equations. In the spirit of recent works due to Berselli [5] and Berselli-Georgiadis [6], it is shown that the energy of weak solutions is invariant if $v\in L^{p}(0,T;B^{\frac1p}_{\frac{2p}{p-1},c(\mathbb{N})} )$ with $1<p\leq3$ and the helicity is conserved if $v\in L^{p}(0,T;B^{\frac2p}_{\frac{2p}{p-1},c(\mathbb{N})} )$ with $2<p\leq3 $ for both the periodic domain and the whole space, which generalizes the classical work of Cheskidov-Constantin-Friedlander-Shvydkoy in [10]. This indicates the role of the time integrability, spatial integrability and differential regularity of the velocity in the conserved quantities of weak solutions of the ideal fluid.