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 $Ω= \Torus^2 \times (-1, 1)$ 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 $u_0\in B_{q,p}^s(\Omega )$ for $s>2-2/p+1/q$ converges to the solution to the primitive equations with the same initial data in ${\mathbb{E}}_1(T)=W^{1,p}(0,T;L^q(\Omega ))\cap L^p(0,T;W^{2,q}(\Omega ))$ with order $O(\epsilon )$ where $(p,q)\in (1,\infty {)}^2$ satisfies $ \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }$. The global well-posedness of the scaled Navier-Stokes equations in ${\mathbb{E}}_1(T)$ is also proved for sufficiently small $\epsilon >0$. Note that $T=\infty$ is included.

24pages