On the Energy Equality via a Priori Bound on the Velocity for Axisymmetric 3D Navier–Stokes Equations

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

$$\begin{aligned} |\tilde{u}|\le \frac{1}{r}\,,\quad 0< r\le 1\,, \end{aligned}$$(0.1)

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

$$\begin{aligned} |\tilde{u}|\le \frac{1}{r^d}\,,\quad 0< r\le 1\,,\quad d>1\,, \end{aligned}$$(0.2)

and

$$\begin{aligned} \nabla \tilde{u}\in L^{\frac{6d-4}{2d-1}}(0,T;L^{2}(\mathbb {R}^3))\,, \end{aligned}$$(0.3)

then the energy equality holds.