Abstract
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.
Highlights
The rigorous justification of the primitive equations from the scaled Navier–Stokes equations was studied by Azérad and Guillén [2]
It is more physically natural to consider the case of Dirichlet–Neumann and Dirichlet boundary conditions, there were no results to justify the derivation of the primitive equations from the Navier–Stokes equations in a strong topology
Lemma 3 follows from a maximal regularity estimate involving the Stokes operator, which follows from a bound for the pure imaginary powers by Dore–Venni theory
Summary
In Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞), Keywords: Hydrostatic approximation, Scaled Navier–Stokes equations, Maximal regularity. Hieber and Kashiwabara [23] extended this result to prove global well-posedness for the primitive equations in L p-settings In these papers, boundary conditions are imposed no-slip (Dirichlet) on the bottom and slip (Neumann) on the top. The rigorous justification of the primitive equations from the scaled Navier–Stokes equations was studied by Azérad and Guillén [2] They obtained weak* convergence in the natural energy space L∞(0, T ; L2(Ω)) ∩ L2(0, T ; H 1(Ω)) for Ω = T2 × (−1, 1) and T > 0. It is more physically natural to consider the case of Dirichlet–Neumann and Dirichlet boundary conditions, there were no results to justify the derivation of the primitive equations from the Navier–Stokes equations in a strong topology. Let u and u be a solution of (PE) and (SNS) in E1(T ) under the Dirichlet boundary condition with initial data u0, respectively, such that.
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