ON THE ENERGY EQUALITY FOR AXISYMMETRIC WEAK SOLUTIONS TO THE 3D NAVIER–STOKES EQUATIONS

Abstract

We focus on 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 to $L^q(0,T;L^p({ℝ}^3))$ with $\frac{1}{q}+\frac{1}{p}=\frac{1}{2}$ and $p\ge 4$, then $u$ must satisfy the energy equality. A novel point is that, for the axisymmetric Navier–Stokes equations, the Shinbrot condition can be relaxed as follows: if $ũ=u^r{e}_r+u^z{e}_z\in L^q(0,T;L^p({ℝ}^3))$ with $\frac{1}{q}+\frac{1}{p}=\frac{1}{2}$ and $p\ge 4$, then $u$ must satisfy the energy equality. Some other interesting results will also be obtained.