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THE HYDROSTATIC APPROXIMATION FOR THE PRIMITIVE EQUATIONS BY THE SCALED NAVIER-STOKES EQUATIONS UNDER THE NO-SLIP BOUNDARY CONDITION

FURUKAWA, KEN GIGA, YOSHIKAZU KASHIWABARA, TAKAHITO 北海道大学

2020.06.17

概要

In this paper we justify the hydrostatic approximation of the primitive equations in the maximal Lp-Lq-setting in the three-dimensional layer domain Ω = T2 × (−1, 1) under the no-slip (Dirichlet) boundary condition in any time interval (0, T ) for T > 0. We show that the solution to the scaled Navier-Stokes equations with Besov initial data u0 ∈ Bs q,p (Ω) for s > 2 − 2/p + 1/q converges to the solution to the primitive equations with the same initial data in E1(T ) = W 1,p(0, T ; Lq(Ω)) ∩ Lp(0, T ; W 2,q(Ω)) with order O(ǫ) where (p, q) ∈ (1, ∞)2 satisfies 1 ≤ min(1 − 1/q, 3/2 − 2/q). The global well-posedness of the scaled Navier-Stokes equations in E1(T ) is also proved for sufficiently small ǫ > 0. Note that T = ∞ is included.

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