A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity

Chemin et al. [M2AN Math. Model. Numer. Anal., 34 (2000), pp. 315-335.] considered the three-dimensional Navier-Stokes equations with vanishing vertical viscosity. Assuming that the initial velocity is square-integrable in the horizontal direction and Hs in the vertical direction, they prove existence of solutions for s > 1/2 and uniqueness of solutions for s > 3/2. Here, we close the gap between existence and uniqueness, proving uniqueness of solutions for s > 1/2. Standard techniques are used.