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The hydrostatic approximation for the primitive equations by the scaled Navier–Stokes equations under the no-slip boundary condition
Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2021-03-11 , DOI: 10.1007/s00028-021-00674-6
Ken Furukawa , Yoshikazu Giga , Takahito Kashiwabara

In this paper, we justify the hydrostatic approximation of the primitive equations in maximal \(L^p\)-\(L^q\)-settings in the three-dimensional layer domain \(\varOmega = \mathbb {T} ^2 \times (-1, 1)\) under the no-slip (Dirichlet) boundary condition in any time interval (0, T) for \(T>0\). We show that the solution to the \(\epsilon \)-scaled Navier–Stokes equations with Besov initial data \(u_0 \in B^{s}_{q,p}(\varOmega )\) for \(s > 2 - 2/p + 1/ q\) converges to the solution to the primitive equations with the same initial data in \(\mathbb {E}_1 (T) = W^{1, p}(0, T ; L^q (\varOmega )) \cap L^p(0, T ; W^{2, q} (\varOmega )) \) with order \(O(\epsilon )\), where \((p,q) \in (1,\infty )^2\) satisfies \( \frac{1}{p} \le \min ( 1 - 1/q, 3/2 - 2/q ) \) and \(\epsilon \) has the length scale. The global well-posedness of the scaled Navier–Stokes equations by \(\epsilon \) in \(\mathbb {E}_1 (T)\) is also proved for sufficiently small \(\epsilon >0\). Note that \(T = \infty \) is included.



中文翻译:

在无滑移边界条件下按比例缩放的Navier–Stokes方程对原始方程的静水力逼近

在本文中,我们证明在最大的原始方程的静水近似\(L ^ P \) - \(L ^ Q \)在三维层域-settings \(\ varOmega = \ mathbb横置^在任何时间间隔(0,T)中对于((T> 0 \)在无滑动(Dirichlet)边界条件下为 2 \ times(-1,1)\)。我们证明了对于((s> ,具有Besov初始数据\(u_0 \ in B ^ {s} _ {q,p}(\ varOmega)\)\(\ epsilon \)比例缩放的Navier–Stokes方程的解。 2-2 / p + 1 / q \)收敛到具有相同初始数据的基本方程组的解\(\ mathbb {E} _1(T)= W ^ {1,p}(0,T; L ^ q(\ varOmega))\ cap L ^ p(0,T; W ^ {2,q}( \ varOmega))\)的顺序为\(O(\ epsilon)\),其中\((p,q)\ in(1,\ infty)^ 2 \)满足\(\ frac {1} {p} \ le \ min(1-1 / q,3/2-2 / q)\)\(\ epsilon \)具有长度比例。经缩放的Navier-Stokes方程中由全局适定性\(\小量\)\(\ mathbb {E} _1(T)\)也证明了足够小\(\小量> 0 \) 。请注意,其中包含\(T = \ infty \)

更新日期:2021-03-11
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