Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equationsThis work was partly supported by the DFG International Research Training Group IRTG 1529 and the JSPS Japanese-German Graduate Externship on Mathematical Fluid Dynamics. The second author is partly supported by JSPS through grant Kiban S (No. 26220702), Kiban A (No. 19H00639), Kaitaku (No. 18H05323), Kiban A (No. 17H01091), Kiban B (No. 16H03948), and the fourth and the last auth

• Published in: Nonlinearity 2020-10, Vol.33 (12), p.6502-6516
• Main Authors: Furukawa, Ken, Giga, Yoshikazu, Hieber, Matthias, Hussein, Amru, Kashiwabara, Takahito, Wrona, Marc
• Format: Article
• Language: English
• Subjects: convergence rate; primitive equations; scaled Navier-Stokes equations; strong convergence
• 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 u0=(v0,w0)∈Bq,p2−2/p, 1/q + 1/p ⩽ 1 if q ⩾ 2 and 4/3q + 2/3p ⩽ 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 v0 with respect to the maximal-Lp-Lq-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 L2-L2-setting. The approach presented here does not rely on second order energy estimates but on maximal Lp-Lq-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.