On the two-dimensional hydrostatic Navier-Stokes equations

This paper concerns some mathematical results on the two-dimensional hydrostatic equations, also called the primitive equations. The uniqueness of weak solutions of the two-dimensional Navier-Stokes equations is well known. Such a result is not known on the two-dimensional hydrostatic equations with Dirichlet boundary condition on the bottom. These equations are derived from the Navier-Stokes equations replacing the vertical component of the momentum equations by the hydrostatic equation on the pressure. We give here some partial answers on the uniqueness, the global strong existence of solutions, and the exponential decay in time of the energy. We assume a basin with a strictly positive depth. The degenerate case, in which the depth vanishes on the shore, remains open.