2024-03-29T00:24:56Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/712872022-11-17T02:08:08Zhdl_2115_45007hdl_2115_116RIGOROUS JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION FOR THE PRIMITIVE EQUATIONS BY SCALED NAVIER-STOKES EQUATIONSFURUKAWA, KENGIGA, YOSHIKAZUHIEBER, MATTHIASHUSSEIN, AMRUKASHIWABARA, TAKAHITOWRONA, MARC410Consider 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.Department of Mathematics, Hokkaido UniversityDepartmental Bulletin Paperapplication/pdfhttp://hdl.handle.net/2115/71287info:doi/10.14943/85422https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/71287/1/RigorousJustification.pdfHokkaido University Preprint Series in Mathematics11121102018-08-10engpublisher