On the energy equality via a priori bound on the velocity for axisymmetric 3D Navier-Stokes equations (Q6564128)

In this paper, the author considered the energy equality for weak solutions of the 3D Navier-Stokes equations. For the axisymmetric weak solution \({\tilde u}=u^r e_r+u^ze_z\), the author proved that the energy equality holds if these weak solution are such that \N\[\N|\tilde u|\le \frac1{r^d}, \;\; 0<r\le 1,\;\; d>1\N\]\Nand \N\[\N\nabla \tilde u\in L^{\frac{6d-4}{2d-1}}(0,T; L^{2}(\mathbb R^3)).\N\]