Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations

Considering the anisotropic Navier-Stokes equations as well as the primitive equations, it is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height ϵ with initial data, 1/q + 1/p 1 if q 2 and 4/3q + 2/3p 1 if q 2, converges as ϵ → 0 with convergence rate to the horizontal velocity of the solution to the primitive equations with initial data v 0 with respect to the maximal-L p -L q -regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the L 2-L 2-setting. The approach presented here does not rely on second order energy estimates but on maximal L p -L q -estimates which allow us to conclude that local in-time convergence already implies global in-time convergence, where moreover the convergence rate is independent of p and q.