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

• Published in: Nonlinearity 2020-12, Vol.33 (12), p.6502-6516
• Main Authors: Furukawa, Ken, Giga, Yoshikazu, Hieber, Matthias, Hussein, Amru, Kashiwabara, Takahito, Wrona, Marc
• Format: Article
• Language: English
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/aba509

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 ɛ with initial data u 0 = ( v 0 , w 0 ) ∈ B q , p 2 − 2 / p , 1/ q + 1/ p ⩽ 1 if q ⩾ 2 and 4/3 q + 2/3 p ⩽ 1 if q ⩽ 2, converges as ɛ → 0 with convergence rate O ( ε ) 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 .