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In this paper, we are concerned with the minimal regularity of weak solutionsimplying the law of balance for both energy and helicity in the incompressibleEuler equations. In the spirit of recent works due to Berselli [5] andBerselli-Georgiadis [6], it is shown that the energy of weak solutions isinvariant 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 $v\inL^{p}(0,T;B^{\frac2p}_{\frac{2p}{p-1},c(\mathbb{N})} )$ with $2<p\leq3 $ forboth the periodic domain and the whole space, which generalizes the classicalwork of Cheskidov-Constantin-Friedlander-Shvydkoy in [10]. This indicates therole of the time integrability, spatial integrability and differentialregularity of the velocity in the conserved quantities of weak solutions of theideal fluid.

On the energy and helicity conservation of the incompressible Euler equations