We consider the Navier--Stokes equations in~$\mathbb R^3$, in an axisymmetric setting: the data and the solutions only depend on the radial and on the vertical variable. In , a unique solution is constructed in a scale invariant function space~$L^2_0$, equivalent to~$L^2$ at finite distance from the vertical axis. We prove here a weak--strong uniqueness result for such solutions associated with data in~$L^2 \cap L^2_0$.
"Stability and weak-strong uniqueness for axisymmetric solutions of the Navier-Stokes equations." Differential Integral Equations 16 (5) 557 - 572, 2003.