论文标题
在无滑动边界条件下缩放的Navier-Stokes方程原始方程的静静力近似
The Hydrostatic Approximation for the Primitive Equations by the Scaled Navier-Stokes Equations under the No-Slip Boundary Condition
论文作者
论文摘要
在本文中，我们证明原始方程在最大$ l^p $ - $ l^p $ - $ l^q $ - 在三维层域$ω= \ torus^2 \ torus^2 \ torus^2 \ terus^2 \ times（-1，1）$在任何时间间隔$（0，t）$ for $ t for $ t for $ t for $ t for $ t for $ t for $ t for $ t for $ t for $ t for $ t for $ t> 0 $ t for No-S-SLIP（Dirichlet）边界条件下。我们表明，使用BESOV的最初数据$ u_0 \ in B^{s} _ {q，p}（ω）$ in b^{s> 2-2-2/ p + 1/ q $在$ s> 2-2/ p）中收敛到$ s> 2-2/ p +的方法，并在$ s> 2-2/ p + \ s> p + s}中收敛到$ s> 2-2/ p + 1/ q， （（ω））\ cap l^p（0，t; w^{2，q}（ω））$，带$ o（ε）$其中$（p，q）\ in（1，\ infty）^2 $满足$ \ frac {1} {p} {p} {p} {p} \ leq \ leq \ leq \ sin \ bracket \ bracket {1-1-1-1-1-2-2-2-2-2-2-/q/q/q/q/q/q/q/q/q/q/q/q/q/q/q/q/q/q/q。 $ \ mathbb {e} _1（t）$中缩放的Navier-Stokes方程的全局良好性也被证明是足够小的$ε> 0 $。请注意，包括$ t = \ infty $。
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 $u_0 \in B^{s}_{q,p}(Ω)$ for $s > 2 - 2/p + 1/ q$ converges to the solution to the primitive equations with the same initial data in $\mathbb{E}_1 (T) = W^{1, p}(0, T ; L^q (Ω)) \cap L^p(0, T ; W^{2, q} (Ω)) $ with order $O(ε)$ 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 $ε>0$. Note that $T = \infty$ is included.
