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 epsilon with initial data u(0)=(v(0),w(0)) is an element of B-q,p(2-2/p), 1/q + 1/p <= 1 if q >= 2 and 4/3q + 2/3p <= 1 if q <= 2, converges as epsilon -> 0 with convergence rate O(epsilon) 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.