In this paper, we are concerned with the energy equality for axisymmetric weak solutions of the 3D Navier–Stokes equations. The classical Shinbrot condition says that if the weak solution u of the Navier–Stokes equations belongs \(L^q(0,T;L^p(\mathbb {R}^3))\) with \(\frac{1}{q}+\frac{1}{p}=\frac{1}{2}\) and \(p\ge 4\) , then u must satisfy the energy equality. For the axisymmetric Navier–Stokes equations, in our previous paper, we found that it is enough to impose the Shinbrot condition to \(\tilde{u}=u^re_r+u^z e_z\) . The recent papers (Chiun-Chuan et al., Commun PDE 34(1–3):203–232, 2009; Koch et al., Acta Math 203(1):83–105, 2009) tell us if 0.1 \(\begin{aligned} |\tilde{u}|\le \frac{1}{r}\,,\quad 0< r\le 1\,, \end{aligned}\) then u is smooth , therefore the energy equality holds. It is natural to ask the relation between a priori bound on the velocity and the energy conservation. The aim of this paper is to investigate this problem. We shall prove that if 0.2 \(\begin{aligned} |\tilde{u}|\le \frac{1}{r^d}\,,\quad 0< r\le 1\,,\quad d>1\,, \end{aligned}\) and 0.3 \(\begin{aligned} \nabla \tilde{u}\in L^{\frac{6d-4}{2d-1}}(0,T;L^{2}(\mathbb {R}^3))\,, \end{aligned}\) then the energy equality holds.
