On the Energy and Helicity Conservation of the Incompressible Euler Equations

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 (J Differ Equ 368:350–375, 2023) and Berselli and Georgiadis (Nonlinear Differ Equ Appl 31(33):1–14, 2024), it is shown that the energy of weak solutions is invariant if the velocity \(v\in L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb {N})} )\) $v\in L^p(0,T;B_{\frac{2p}{p-1},c(\mathbb{N})}^{\frac{1}{p}})$ with \(1<p\le 3\) $1<p\le 3$ and the helicity is conserved if \(v\in L^{p}(0,T;B^{\frac{2}{p}}_{\frac{2p}{p-1},c(\mathbb {N})} )\) $v\in L^p(0,T;B_{\frac{2p}{p-1},c(\mathbb{N})}^{\frac{2}{p}})$ with \(2<p\le 3 \) $2<p\le 3$ for both the periodic domain and the whole space, which generalizes the classical work of Cheskidov et al. (Nonlinearity 21:1233–1252, 2008). As an application, we deduce the upper bound of energy dissipation rate of the form \(o(\mu ^{\frac{p\alpha -1}{p\alpha -2\alpha +1}})\) $o({\mu }^{\frac{p\alpha -1}{p\alpha -2\alpha +1}})$ of Leray–Hopf weak solutions in \(L^{p}( 0,T;\underline{B}^{\alpha }_{\frac{2p}{p-1},VMO}(\mathbb {T}^{d}))\) $L^p(0,T;{\underline{B}}_{\frac{2p}{p-1},VMO}^{\alpha }({\mathbb{T}}^d))$ in the Navier–Stokes equations, which extends recent corresponding result obtained by Drivas and Eyink (Nonlinearity 32:4465–4482, 2019).