open access

https://doi.org/10.14943/85422

https://hdl.handle.net/2115/71287

RIGOROUS JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION FOR THE PRIMITIVE EQUATIONS BY SCALED NAVIER-STOKES EQUATIONS

英語

2018-08-10

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1112

1-10

Consider 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 u0 = (v0,w0) ∈ B2−2/p q,p , 1/q + 1/p ≤ 1 if q ≥ 2 and 4/3q+2/3p ≤ 1 if q ≤ 2, converges as ε → 0 with convergence rateO(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 for the heat equation.

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