The Hydrostatic Approximation for the Primitive Equations by the Scaled Navier-Stokes Equations under the No-Slip Boundary Condition

Abstract

In this paper we justify the hydrostatic approximation of the primitive equations in the maximal L p -L q -setting in the three-dimensional layer domain Ω = T 2 × (−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 u 0 ∈ B s q,p (Ω) for s > 2 − 2/p + 1/q converges to the solution to the primitive equations with the same initial data in E 1 (T ) = W 1,p (0, T ; L q (Ω)) ∩ L p (0, T ; W 2,q (Ω)) with order O(ǫ) where (p, q) ∈ (1, ∞) 2 satisfies 1 p ≤ min(1 − 1/q, 3/2 − 2/q). The global well-posedness of the scaled Navier-Stokes equations in E 1 (T ) is also proved for sufficiently small ǫ > 0. Note that T = ∞ is included.

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

“…In fact, our result can be extended to the problem with the no-slip boundary condition on top and bottom, cf. [5], rather than the periodic boundary condition. Also, we may choose μ z in (1.2) as μ z = δε 2 for some δ > 0 to obtain the anisotropic Navier-Stokes equations as in (1.3), but now with v ε,δ and w ε,δ and with Δ being replaced by Δ H + δ∂ 2 z .…”

The primitive equations approximation of the anisotropic horizontally viscous Navier-Stokes equations

“…As already mentioned at the beginning of this introduction, the rigorous justifications of the limiting process in the case α = 2, i.e., the convergence from (1.4) with α = 2 to (1.9) has been established by Azérad-Guillén [1] in the weak setting and by Li-Titi [38] in the strong setting, respectively, see also Furukawa et al [16] and [17] for some generalizations in the L p -L q type spaces. To our best knowledge, the corresponding justification in the case α > 2, i.e., the convergence from from (1.4) with α > 2 to (1.10), is still unknown, and we are going to address this problem in the current paper.…”

On the hydrostatic approximation of compressible anisotropic Navier-Stokes equations -- rigorous justification

“…Li and Titi [38] used the method of weakstrong uniqueness to prove the aspect ratio limit of incompressible anisotropic Navier-Stokes equations, that is from weak solutions of anisotropic Navier-Stokes equations to strong solutions of incompressible PE model. Then Giga, Hieber and Kashiwabara et al [26,27] extended the results into maximal regularity spaces. Recently, Donatella and Nora [17] proved the convergence in downwind-matching coordinates.…”