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.
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1 Introduction
1.1 Problems and main results
We consider the primitive equations of the form
where \(u = (v, w) \in \mathbb {R}^{2} \times \mathbb {R} \) and \(\pi \) are the unknown velocity field and pressure field, respectively, \(\nabla _H = ( \partial _x, \partial _y ) ^T\), and \(\varOmega = \mathbb {T} ^2 \times (-1, 1)\) for \( \mathbb {T} = \mathbb {R} / 2 \pi \mathbb {Z}\). By the divergence-free condition, w is given by the formula
here, we invoked physically reasonable condition \(w (\cdot , \cdot , \pm 1 , \cdot ) = 0\). The primitive equations are a fundamental model for geographical flow. Existence of a global weak solution to the primitive equations on the spherical shell with thickness \(a > 0\) for \(L^2\)-initial data was proved by Lions, Temam and Wang [29]. Local-in-time well-posedness was proved by Guillén-González, Masmoudi and Rodríguez-Bellido [21]. Although the global well-posedness of the three-dimensional Navier–Stokes equations is the well-known open problem, this problem has been solved by Cao and Titi [3]. 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. Recently, the second and last authors together with Gries, Hieber and Hussein [15] obtained global-in-time well-posedness in maximal regularity spaces (mixed Lebesgue–Sobolev spaces) \( W^{1,p} ( 0, T ; L^q ( \varOmega ) ) \cap L^p ( 0, T ; W^{2,q} ( \varOmega ) ) \) for \(T>0\) and appropriate \(1< p,q < \infty \) under the Neumann, Dirichlet and Dirichlet–Neumann mixed boundary conditions.
Our aim in this paper is to give a rigorous justification of the derivation of the primitive equations under the Dirichlet boundary condition. We begin by explaining its derivation. Let us consider the anisotropic viscous Navier–Stokes equations in a thin domain of the form
where \(\varOmega _{\epsilon } = (- \epsilon , \epsilon ) \times \mathbb {T}^2\). If \(\epsilon = 1\), (ANS) is the usual Navier–Stokes equations.
The equations (ANS) are considered as a good model to describe the motion of an incompressible viscous fluid filled in a thin domain. Actually, if we put the Reynolds number to be equal to one, the apparent viscosity for the vertical direction must be of \(\epsilon ^2\)-order since the vertical length and velocity are of \(\epsilon \)-order. The primitive equations are formally derived from (ANS). We introduce new unknowns from the solution to (ANS) such that
where \(x, y \in \mathbb {T} \), \(z \in (-1, 1)\) and \(t> 0\). Then, \((u_{\epsilon }, \pi _{\epsilon })\) satisfy the scaled Navier–Stokes equations in the fixed domain
Taking formally \(\epsilon \rightarrow 0\) for the above equations, we get the primitive equations.
The Navier–Stokes equations (SNS) are well studied for \(\epsilon = 1\) since the work of Leray [27]. In this paper, he considered a global weak solution in \(\varOmega = \mathbb {R} ^3\). For a general domain, see Farwig et al. [7]. Fujita and Kato [9] constructed local strong solution for \(H^{1/2}\)-initial data. This result was extended to various domains and various function spaces; see, e.g., Ladyzenskaya [25], Kato [24], Giga and Miyakawa [17] for early developments. The reader is referred to the book of Lemarié-Rieusset [26] and review articles by Farwig, Kozono and Sohr [8] and Gallagher [13] for recent developments. Many results can be extended for general \(\epsilon > 0\), but it is not often written explicitly except in a book of Chemin, Desjardins, Gallagher and Grenier [4].
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 \(L^{\infty }( 0, T; L^2(\varOmega )) \cap L^2( 0, T; H^1(\varOmega )) \) for \(\varOmega = \mathbb {T} ^2 \times (-1, 1)\) and \(T>0\). Recently, Li and Titi [28] improved their result to get strong convergence by energy method with the aid of regularity of the solution to the primitive equations. The authors together with Hieber, Hussein and Wrona [11] extended Li and Titi’s result in maximal regularity spaces \(W^{1, p} (0, T ; L^q ( \mathbb {T} ^3)) \cap L^{p} (0, T ; W^{2, q} ( \mathbb {T} ^3))\) with initial trace in the Besov space \(B^{2 - 2 / p}_{q,p} {( \mathbb {T} ^3)}\) for \(T > 0\) and \(1 / p \le \min (1 - 1 / q, 3 / 2 - 2 / q)\) by an operator theoretic approach. The case of \(p = q = 2\) is corresponding to Li and Titi’s result. Note that the case of the torus corresponds to the Neumann boundary conditions on the top and bottom parts. In the work of Azérad and Guillén, they considered the case of mixed boundary conditions with the Dirichlet boundary condition on the bottom, while Li and Titi treated the case of the Neumann boundary condition only. As we already mentioned, the primitive equations are a model for geophysical flow. Although 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
and
be the initial trace space of \(\mathbb {E}_1 (T)\), where \(B^s_{q,p} {(\varOmega )}\) denotes the \(L^q\)-Besov space of order s. In this paper, we frequently use \(\Vert \cdot \Vert _{\mathbb {E}_0 (T)}\) as the norm of \(L^p (0, T ; L^q (\varOmega ))\) and \(\Vert \cdot \Vert _{\mathbb {E}_1 (T)}\) as the norm of \(W^{1,p} (0, T ;L^q(\varOmega )) \cap L^p (0, T; W^{2,q}(\varOmega ))\) to simplify the notation.
Let us seek the solution \(U_\epsilon = (V_\epsilon , W_\epsilon )\) to
where
The system (1) is the equations of the difference between solutions to (PE) and (SNS).
Theorem 1
Let \(T > 0\). Suppose \((p,q) \in (1, \infty )^2\) satisfies \( \frac{1}{p} \le \min ( 1 - 1/q, \, 3/2 - 2/q ) \), \(u_0 = (v_0, w_0) \in X_{\gamma }\) and \(v_0 \in B^{s}_{q, p} (\varOmega )\) for \(s > 2 - 2 / p + 1 / q\). Let \(u \in \mathbb {E}_1 (T)\) be a solution of (PE) with initial data \(u_0\). Then, there exist positive constants \(\epsilon _0 = \epsilon _0 (p, q, \Vert u \Vert _{\mathbb {E}_1 (T)})\), \(C = C (p, q, \Vert u \Vert _{\mathbb {E}_1 (T)})\) and a unique solution \(U_\epsilon = (V_\epsilon , W_\epsilon )\) to (1) such that
for any \(\epsilon \in (0, \epsilon _0)\). Moreover, \(u_\epsilon = (v_\epsilon , w_\epsilon ) := (v + V_\epsilon , w + W_\epsilon )\) is the unique solution to (SNS) in \(\mathbb {E}_1 (T)\).
Note that the case \(T = \infty \) is included. We use the assumption \(v_0 \in B^{s}_{q, p} (\varOmega )\) in Theorem 1 to estimate \(\Vert u \Vert _{\mathbb {E}_1 (T)}\). We give an explicit form of \(\epsilon _0\) in Remark 33. This theorem implies the justification of the hydrostatic approximation.
Corollary 2
Let \(T > 0\) and \(0 < \epsilon \le 1\). Suppose \((p,q) \in (1, \infty )^2\) satisfies \( \frac{1}{p} \le \min ( 1 - 1/q, 3/2 - 2/q ) \), \(u_0 = (v_0, w_0) \in X_{\gamma }\) and \(v_0 \in B^{s}_{q, p} (\varOmega )\) for \(s > 2 - 2 / p + 1 / q\). Let u and \( {u}_{\epsilon } \) be a solution of (PE) and (SNS) in \(\mathbb {E}_1 (T)\) under the Dirichlet boundary condition with initial data \(u_0\), respectively, such that
for some \(C_{0} = C_{0} ( u_0, p, q ) \). Then, there exists a positive \( C = C(p, q, C_0 ) \) such that
1.2 Strategy
Our strategy to show Theorem 1 is based on estimates for \((V_\epsilon , \epsilon W_\epsilon )\). The proof consists of two key steps: maximal regularity results of the anisotropic Stokes operator and the improved regularity result for the vertical component of the solution to the primitive equations. We consider the nonlinear terms in (SNS) as an external force f and set \(u_\epsilon = (v_\epsilon , \epsilon w_\epsilon )\) to get the linear equation
where \(\nabla _\epsilon = (\partial _1, \partial _2 , \partial _3 / \epsilon )^T\) and \(\mathrm {div}_\epsilon = \nabla _\epsilon \cdot \). We define the function space \(\mathbb {E}_{\epsilon , j} (T)\) for \(j = 0, 1\) similarly as \(\mathbb {E}_j (T)\) by replacing \(\mathrm {div}\) by \(\mathrm {div_\epsilon }\). Although the space \(\mathbb {E}_{\epsilon , j} (T)\) depends on \(\epsilon \), the norm is just the norm in \(W^{1,p} (0, T ;L^q(\varOmega )) \cap L^p (0, T; W^{2,q}(\varOmega ))\), so we shall write the norm in \(\mathbb {E}_{\epsilon , j} (T)\) simply by \(\Vert \cdot \Vert _{\mathbb {E}_j (T)}\).
We recall some known results on maximal regularity for the Stokes operator, which is corresponding to the case \(\epsilon = 1\). Solonnikov [35] first proved \( L^{ q } \)-\( L^{ q } \) maximal regularity for the Stokes operator by a potential-theoretic approach. The second author [14] established a bound for the pure imaginary powers of the Stokes operator in a bounded domain; see [18] for an exterior domain. This type of property will be simply called bounded imaginary powers, shortly BIP. This BIP implies the maximal regularity \( L^{ p } \)-\( L^{ q } \) regularity via Dore–Venni theory [6]. Indeed, the second author and Sohr [19] established the maximal regularity in an exterior domain by estimating BIP. Further studies on maximal regularity were done by many researchers, for instance, Dore and Venni [6] and Weis [36]. See Denk et al. [5] for further comprehensive research. In our case, we have to clarify \(\epsilon \)-dependence in estimates for maximal regularity, which is a key point. Our key maximal regularity result is
Lemma 3
Let \(1< p, q < \infty \), \(0 < \epsilon \le 1\) and \(T > 0\). Let \( f \in \mathbb {E}_{\epsilon , 0} (T)\) and \(u_0 \in B^{2 (1 - 1 / p)}_{q, p} (\varOmega )\) with \(\mathrm {div}_\epsilon \, u = 0\). Then, there exist constants \(C = C (p,q)>0\) and \(C^\prime = C^\prime (p,q) > 0\), which are independent of \(\epsilon \), and \((u, \pi )\) satisfying (4) such that
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. However, we need to clarify that C and \(C^\prime \) can be taken independent of \(\epsilon \). For \(\epsilon = 1\), a necessary BIP estimate for the Stokes operator has been established by Abels [1], where a resolvent decomposition similar to [14] is used. Unfortunately, the \(\epsilon \)-dependent case is not discussed there. However, the strategy in [1] works for our problem. We construct the anisotropic Stokes operator by the method in [1] and show the boundedness of its imaginary powers. Note that, in our previous paper [11], maximal regularity of the anisotropic Stokes operator is much easier since the corresponding Stokes operator is essentially the same as the Laplace operator on \( \mathbb {T} ^3\). In the case of the Dirichlet boundary condition, the corresponding Stokes operator becomes to be much more difficult by the effect of boundaries, which is substantially different from the case of the periodic boundary conditions. The maximal regularity was proved in a layer domain for the Stokes operator under various boundary conditions by Saito [33] by proving \(\mathcal {R}\)-boundedness of the resolvent operator when \(\epsilon =1\). Unfortunately, it seems very difficult to check the dependence of \(\epsilon \), so we do not take this approach.
The term \(F = \partial _t w - \varDelta w + u \cdot \nabla w\) appears in the right-hand side of (1). Thus, we need to improve the regularity of w and estimate this term in \(L^p (0, T ; L^q (\varOmega ))\).
Lemma 4
Let \(T > 0\) and \(u_0 = (v_0, w_0) \in X_\gamma \) with \(w_0 = - \int _{-1}^{x_3} \mathrm {div}_H \, v_0 \ \mathrm{d} \zeta \) and \(v_0 \in B^{s}_{q, p}(\varOmega )\) for \(s > 2 - 2 / p + 1 / q\) and \(u = (v, w)\) be the solution to (PE). Assume \(v \in \mathbb {E}_1 (T)\). Then, there exists a constant \(C> 0\) such that
Since it is known that \(v \in \mathbb {E}_1 (T)\), as we have already mentioned, we see \(w (\cdot , x_3) = - \int _{-1}^{x_3} \mathrm {div}_H v (\cdot , \zeta ) \mathrm{d}\zeta \in W^{1, p}( 0, T ; W^{-1, q} (\varOmega )) \cap L^p (0, T; W^{1, p} (\varOmega ))\). This derivative loss is due to the absence of the equation of time-evolution of w in the primitive equations. In our previous paper [11], which treats the periodic boundary condition, we recover the regularity of w by deriving the equation which w satisfies and applying maximal regularity of the Laplace operator to the equation. However, in the case of the Dirichlet boundary condition, this method is not applicable directly because of the presence of the second-order derivative term at the boundary, which vanishes in the case of periodic boundary condition. Thus, we are forced to impose additional regularity for initial data to get regularity for v. If \(v_0 \in B^{s}_{q,p} (\varOmega )\) for \(s > 2 - 2 /p + 1 /q\), then we obtain \(v \in L^p (0 , T; W^{s + 2 / p , q} (\varOmega ))\) and the trace of the second derivative belongs to \(\mathbb {E}_0 (T)\).
Let us explain our strategy to show Theorem 1.
Once Lemma 4 is proved, our main result Theorem 1 can be proved essentially the same way as [11], where the proof we give here is slightly different from that of [11]. Moreover, we give an explicit form of the constant C in Theorem 1; see the estimate (96). We first show the boundedness of nonlinear terms \(F_H\) and \(F_z\) in (1) in the space \(\mathbb {E}_0 (T)\). We know that F is also bounded in \(\mathbb {E}_0 (T)\) by Lemma 3. We next apply Lemma 4 to (1) to get a quadratic inequality, which leads to \( { \left| \left| ( {V}_{\epsilon } , \epsilon {W}_{\epsilon } ) \right| \right| }_{\mathbb {E}_1 ( T^{*} ) } \le C \epsilon \) for some short time \(T^{*}> 0\) and \(\epsilon \)-independent constant \(C>0\). Since C depends only on p, q, \(u_0\), \(\Vert u \Vert _{\mathbb {E}_1 (T)}\) and T, if we take \(\epsilon \) small, we are able to extend the time to all finite time T by finite step. In the case of \(T = \infty \), we show (2) for sufficiently large \(T^\prime > 0\). We extend the existence time of \(U_\epsilon \) from \([0, T^\prime )\) to \([0, \infty )\) by the above procedure and smallness of \(\Vert u \Vert _{\mathbb {E}_1 (\infty )} - \Vert u \Vert _{\mathbb {E}_1 (T^\prime )}\).
This paper is organized as follows. In Sect. 2, the boundedness of pure imaginary powers is proved. The resolvent operator of the anisotropic Stokes operator is decomposed into three parts, and each part of uniform bound independent of \(\epsilon \) is proved. In Sect. 3, improved regularity for w is proved. In Sect. 4, we give a proof of our main theorem by iteration.
1.3 Notation
We introduce our notation. We denote by \(|| \cdot ||_{X \rightarrow Y}\) the operator norm from a Banach space X to a Banach space Y. We denote by \(C_0^\infty (\varOmega )\) the set of compactly supported smooth functions in \(\varOmega \). We write \(L^q (\varOmega )\) to denote the Lebesgue space for \(1 \le q \le \infty \) equipped with the norm
We use the usual modification when \(q = \infty \). For a three-dimensional \(L^q\)-vector field f on a domain D, we write \(f \in L^q (D)\) to simplify the notation. For \(m \in \mathbb {Z}_{\ge 0}\) and \(1 \le q \le \infty \) we denote by \(W^{m, q} (\varOmega )\) the m-th-order Sobolev space equipped with the norm
We define the fractional Sobolev spaces \(W^{s, q} (\varOmega ) (= B^s_{qq} (\varOmega ))\) for \(s \notin \mathbb {Z}\) and \(1< q < \infty \) by the real interpolation \(\left( W^{[s], q} (\varOmega ), W^{[s] + 1, q} (\varOmega ) \right) _{s - [s], q}\), where \([\cdot ]\) denotes the Gauss symbol. We define the Bessel potential spaces \(H^{s, q}\) by the complex interpolation \(\left[ W^{s, p}, W^{[s] + 1, q} \right] _{s - [s]}\) for \(s \in \mathbb {R} _+ {\setminus } \mathbb {Z}_{\ge 1}\) and \(1< q < \infty \). We define the Fourier transform by
and the Fourier inverse transform by
The Fourier transform on the d-dimensional torus \( \mathbb {T} ^d\) and its inverse transform are defined by \([ \mathcal {F}_d f ] (n) = {\widehat{f}_n} = \int _{ \mathbb {T} ^d} e^{ - i x \cdot n} f (x) \mathrm{d}x\) and \([ \mathcal {F}_d^{-1} g ] (x) = \frac{1}{(2 \pi )^d} \sum _{n} g_n e^{i n \cdot x}\), respectively. We denote by \( \mathcal {F}_{ {x}^{\prime } }\) the partial Fourier transform with respect to \( {x}^{\prime } \in \mathbb {R} ^2\) and by \( \mathcal {F}^{ - 1 } _{ {\xi }^{\prime } }\) the partial Fourier inverse transform with respect to \( {\xi }^{\prime } \). We denote by \( \mathcal {F}_{d, {x}^{\prime } }\) the partial Fourier transform with respect to \( {x}^{\prime } \in \mathbb {T} ^2\) and by \( \mathcal {F}^{ - 1 } _{d, {n}^{\prime } }\) the partial Fourier inverse transform with respect to \( {n}^{\prime } \in \mathbb {Z}^2\). We write \(\varSigma _{\theta } := \left\{ \lambda \in \mathbb {C} : \left| \arg \lambda \right| < \pi - \theta \right\} \). For a Fourier multiplier operator \( \mathcal {F}^{ - 1 } _{\xi } m ( \xi ) \mathcal {F}_x\) in \( \mathbb {R} ^3\), we denote by \([m]_{\mathcal {M}}\) the Mikhlin constant. \( \mathcal {F}^{ - 1 } _{ {\xi }^{\prime } } m ( \xi ) \mathcal {F}_{x^\prime }\) is a Fourier multiplier operator in \( \mathbb {R} ^2\) with Mikhlin constant \([m]_{ {\mathcal {M}}^{\prime } }\). For \(0 < \epsilon \le 1\), \( {\varDelta }_{\epsilon } = \partial _1^2 + \partial _2^2 + \partial _3^2 / \epsilon ^2\) denotes the anisotropic Laplace operator. We denote by \(E_0\) the zero-extension operator with respect to the vertical variable from \((-1 , 1)\) to \( \mathbb {R} \), i.e.,
for a function f defined in \( \mathbb {R} ^2 \times (-1, 1)\) or \( \mathbb {T} ^2 \times (-1, 1)\). We denote by \(R_0\) the restriction operator with respect to the vertical variable from \( \mathbb {R} \) to \((-1, 1)\). For an integrable function f defined in \(\varOmega \), we write its vertical and horizontal average by \(\overline{f} = \frac{1}{2} \int _{-1}^{1} f (\cdot , \cdot , \zeta ) \ \mathrm{d} \zeta \) and \(\mathrm {ave}_H (f) = \int _{ \mathbb {T} ^2} f (x^\prime , \cdot )\mathrm{d}x^\prime \), respectively.
2 A uniform bound for pure imaginary powers of the anisotropic Stokes operator and its maximal regularity
In this section, we first establish a uniform bound independent of \(\epsilon \) for the pure imaginary powers to the anisotropic Stokes operator along with [1]. Then, we shall give the proof of Lemma 3.
2.1 Boundedness of Fourier multipliers
Although the case of the infinite layer \( \mathbb {R} ^2 \times (-1, 1)\) is considered in [1], his method also works in the case of the periodic layer \(\varOmega = \mathbb {T} ^2 \times (- 1, 1)\) thanks to Fourier multiplier theorem on the torus, e.g., Proposition 4.5 in [22] and Sect. 4 of Grafakos’s book [20].
Proposition 5
[22] Let \(1< p < \infty \) and \(m \in C^{d} ( \mathbb {R} ^d {\setminus } \{ 0 \})\) satisfies the Mikhlin condition:
Let \(a_k = m (k)\) for \(k \in \mathbb {Z}^d {\setminus } \{ 0 \}\) and \(a_0 \in \mathbb {C} \).
For \(f (x) = \sum _{n \in \mathbb {Z}^d} \widehat{f}_n e^{i n \cdot x} \in L^{q} ( \mathbb {T} ^d)\) and a sequence \(a = \{ a_n \}_{n \in \mathbb {Z}^d}\), we set the Fourier multiplier operator of discrete type by
Then, there exists a constant \(C = C(p,d) > 0\) such that
Let us consider the resolvent problem to (4);
for \(\lambda \in \varSigma _\theta \) (\(0< \theta < \pi / 2\)) and \(f \in L^{ q } ( \varOmega ) \). As in the case \(\epsilon = 1\), there is a topological direct sum decomposition called the (anisotropic) Helmholtz decomposition:
for \(q \in (1, \infty )\); see Miyakawa [30] for infinite layer domain. The reader is referred to the book of Galdi [12] for fundamentals of the Helmholtz projection. Let \(H_\epsilon \) be the projection from \(L^q (\varOmega )\) to \( L^{q}_{\sigma , \epsilon } (\varOmega )\) associated with this decomposition. We shall prove its \(L^q\)-boundedness in Lemma 16. This projection is called the anisotropic Helmholtz projection. Let \(A_\epsilon = H_\epsilon (- \varDelta )\) be the Stokes operator with the domain \(D (A_\epsilon ) = L^{q}_{\sigma , \epsilon } (\varOmega ) \cap W^{2, q} (\varOmega )\). For \(0< a < 1 / 2\) and \( - a< \mathrm {Re} \, z < 0\), the fractional power of \(A_{\epsilon }\) is defined via the Dunford calculus
where \( 0< \theta < \pi /2 \) and \(\varGamma _{\theta } = \mathbb {R} e ^{ i ( - \pi + \theta ) } \cup \mathbb {R} e ^{ i ( \pi - \theta ) } \). Our aim in this section is to prove
Lemma 6
Let \(1< q < \infty \), \(0 < \epsilon \le 1\), \( 0< a < 1/2\), \(z \in \mathbb {C} \) satisfying \( - a< \mathrm {Re} \, z < 0 \) and \( 0< \theta < \pi /2 \). Then, there exists a constant \(C = C ( q, a, \theta ) \) such that
Once the above lemma is proved, then we obtain the maximal regularity of the anisotropic Stokes operator via the formula
for \(0< c < 1\) and the Dore–Venni theory [6].
To show Lemma 6, we decompose the solution \((u, \pi )\) to (10) into three parts:
where \(v_j\) and \(\pi _j\) are solutions to
and
where \(\nu \) is the unit outer normal and \(\gamma = \gamma _{\pm }\) is the trace operator to the upper and lower boundary, namely,
for a vector field f on \( \mathbb {T} ^2 \times \mathbb {R} \) or \( \mathbb {R} ^3\). To show Lemma 6, we need to obtain the estimate
with some constant \(C >0\) independent of \(\epsilon \), z and f.
Remark 7
For \(f \in L^q (\varOmega )\) and its horizontal average \(\mathrm {ave}_H (f) = \int _{ \mathbb {T} ^2} f \mathrm{d} x^\prime \), we solve the resolvent problem (10) with external force \(\mathrm {ave}_H (f)\) to get \(u = \left( (\lambda - \partial _3^2)^{-1} \mathrm {ave}_H (f_H) , 0 \right) ^T\) and \(\pi = \epsilon \int _{-1}^{x_3} \mathrm {ave}_H (f_3) \mathrm{d} \zeta / \mathbb {R}\), where \(f_H\) is the horizontal component of f and \( / \mathbb {R}\) means average-free. Since \(- \partial _3^2\) has BIP and the resolvent operator is linear, by taking the difference between the solution to (10) and \((u, \pi )\), we can always assume without loss of generality that f is horizontal average-free.
We define the space of horizontally average-free \(L^q\)-vector fields by
Similarly, we define
Throughout this section, we frequently use partial Fourier transforms to construct solutions and estimate these partial Fourier multipliers.
Proposition 8
[1] Let \(1< q < \infty \) and \(a, \, b \in \{-1 , 1 \}\). Let an integral operator M be
for \(f \in L^q (\varOmega )\). Then, there exists a constant \(C>0\) such that
Proof
See Lemma 3.3 in [1]. \(\square \)
Rescaled \(L^q\)-Fourier multipliers are also bounded \(L^q\) multiplier as a direct consequence of the Mikhlin theorem.
Proposition 9
Let \(1< q < \infty \) and \(0 < \epsilon \le 1\). Let \(m \in C^d ( \mathbb {R} ^d {\setminus } \{ 0 \})\) be a \(L^q\)-Fourier multiplier with the Mikhlin constant \([m]_\mathcal {M} \le C\) for some \(C>0\). Then, the rescaled one \(m_\epsilon (\xi ) := m (\epsilon \xi )\) is also bounded from \(L^q\) into itself such that
Proof
By the Mikhlin theorem, it suffices to prove (15). In fact, we observe that \([m]_{\mathcal {M}} = [m_\epsilon ]_{\mathcal {M}}\) since
\(\square \)
The above proposition is frequently used in this section to get \(\epsilon \)-independent estimate for scaled multipliers. We show boundedness of some Fourier multiplier operators in advance. We set
for \( {\xi }^{\prime } \in \mathbb {R}^2\). In this paper, we use \(s_\lambda \) to denote \((\lambda + \left| n^\prime \right| ^2)^{1/2}\) for \(n^\prime \in \mathbb {Z}^2\) to simplify notation.
Proposition 10
-
1.
Let \(0< \theta < \pi /2\), \(\lambda \in \varSigma _{\theta }\), \(t > 0\) and \(\alpha \) be a positive integer. Then, there exist constants \(c > 0\) and \(C>0\) such that
$$\begin{aligned} \left[ \left| \xi ^{\prime }\right| ^{\alpha } e ^{- t s_{\lambda } } \right] _{ \mathcal {M}^{\prime } } \le C \frac{ e ^{{ - c t \left| \lambda \right| }^{1/2} } }{ t^{\alpha } }, \quad \left[ \frac{ e ^{ - s_{\lambda } } }{ s_{\lambda } } \right] _{\mathcal {M}^{\prime }} \le C \left| \lambda \right| ^{ - 1/2 } e ^{ - c \left| \lambda \right| ^{1/2} }. \end{aligned}$$(16) -
2.
Let \( -1 \le x_3 \le 1 \). Then, there exists a constant \(C>0\) which is independent of \(\epsilon \), such that
$$\begin{aligned} \left[ \frac{ \sinh ( \epsilon \left| \xi ^{\prime }\right| x_3 ) }{ \sinh ( \epsilon \left| \xi ^{\prime }\right| ) } \frac{ \epsilon \left| \xi ^{\prime }\right| }{ 1 + \epsilon \left| \xi ^{\prime }\right| } \right] _{\mathcal {M}^{\prime }} \le C, \quad \left[ \frac{ \cosh ( \epsilon \left| \xi ^{\prime }\right| x_3 ) }{ \sinh ( \epsilon \left| \xi ^{\prime }\right| ) } \frac{ \epsilon \left| \xi ^{\prime }\right| }{ 1 + \epsilon \left| \xi ^{\prime }\right| } \right] _{\mathcal {M}^{\prime }} \le C \end{aligned}$$(17)for all \(0 < \epsilon \le 1\).
-
3.
Let \(-1 \le x_3 \le 1\). Then, there exists a constant \(C>0\), which is independent of \(\epsilon \), such that
$$\begin{aligned} \left[ \frac{ \sinh ( \epsilon \left| \xi ^{\prime }\right| x_3 ) }{ \cosh ( \epsilon \left| \xi ^{\prime }\right| ) } \right] _{\mathcal {M}^{\prime }} \le C, \quad \left[ \frac{ \sinh ( \epsilon \left| \xi ^{\prime }\right| x_3 ) }{ \cosh ( \epsilon \left| \xi ^{\prime }\right| ) } \right] _{\mathcal {M}^{\prime }} \le C \end{aligned}$$(18)for all \(0 < \epsilon \le 1\).
Proof
The estimate (16) is a direct consequence of Lemma 3.5 in [1] and the Mikhlin theorem.
By definition of \(\sinh \) and \(\cosh \), we find the formula
and
Thus, multiplying both sides of (19) by \(\frac{ \epsilon \left| {\xi }^{\prime } \right| }{ 1 + \epsilon \left| {\xi }^{\prime } \right| }\), we find from Proposition 9 that
The second inequality of (17) is proved by the same way as above using (20). Similarly, by definition of \(\sinh \) and \(\cosh \), the estimate (18) follows. \(\square \)
2.2 Estimate for \(v_1\)
Let us consider the equations (I). For \(a \in \mathbb {R} \) we denote the dilation operator in the last variable by
The anisotropic Helmholtz projection \(\mathbb {P}_{\epsilon }^{ \mathbb {R} ^3}\) on \( \mathbb {R} ^3\) with symbols
is bounded in \(L^q( \mathbb {R} ^3)\) by boundedness of the Riesz operator and the formula
Indeed, it follows from (22) that
since \(\mathbb {P}_1^{ \mathbb {R} ^3}\) is \(L^q\)-bounded, which follows from the boundedness of the Riesz operator. We define the anisotropic Helmholtz projection \(\mathbb {P}_\epsilon ^{ \mathbb {T} ^2 \times \mathbb {R} }\) on \( \mathbb {T} ^2 \times \mathbb {R} \) with symbols by
We find \(\mathbb {P}_\epsilon ^{ \mathbb {T} ^2 \times \mathbb {R} }\) is bounded from \(L^q ( \mathbb {T} ^2 \times \mathbb {R} )\) into itself by boundedness of \(\mathbb {P}_\epsilon ^{ \mathbb {R} ^3}\) and Proposition 5 uniformly in \(\epsilon \in (-1, 1)\).
Proposition 11
Let \( 1< q < \infty \), \(0< a < 1/2\), \(0<\epsilon \le 1\), \(z \in \mathbb {C} \) satisfying \( - a< \mathrm {Re} z < 0\) and \(0< \theta < \pi / 2\). Then, there exists a constant \(C = C ( q, a, \theta ) \) such that
for all \( f \in L^{ q } ( \varOmega ) \).
Proof
It is easy to prove that the Laplace operator on a cylinder \( \mathbb {T} ^2 \times \mathbb {R} \) has BIP by calculating the pure imaginary power directly, see, e.g., Appendix A in [19] and Sect. 8 of Nau [31]. For the Laplace operator, BIP was established on a domain with boundary long time ago by Fujiwara [10] and Seeley [34]. Combining these facts with \(L^q\)-boundedness of \(\mathbb {P}_\epsilon ^{ \mathbb {T} ^2 \times \mathbb {R} }\), we have (23). \(\square \)
Proposition 12
Let \(1< q < \infty \), \(0 < \epsilon \le 1\), \(0< \theta < \pi /2\) and \(\lambda \in \varSigma _\theta \). Then, there exists a constant \(C = C(q) > 0\), which is independent of \(\epsilon \), such that
for all \(f \in L^q (\varOmega )\).
Proof
This follows from Propositions 5 and 9 since \(P_\epsilon ^{ \mathbb {T} ^2 \times \mathbb {R} }\) is uniformly bounded from \(L^q ( \mathbb {T} ^2 \times \mathbb {R} )\) into itself. \(\square \)
Let us calculate the partial Fourier transform for \(v_1\) with respect to the horizontal variable. This is needed to obtain representation formula for \(v_2\) later.
Let \(g \in L^q ( \mathbb {T} ^2 \times \mathbb {R} )\). The solution \(\tilde{v}\) to the equation
is given by
Moreover,
where \(\xi _\epsilon = (n^\prime , \xi _3 / \epsilon ) \in \mathbb {Z}^2 \times \mathbb {R} \) and
The kernel function \(k_{\lambda , \epsilon } ( n^\prime , x_3 ) \) is calculated by the residue theorem. Actually, since poles of \( { \left( \lambda + \left| n^\prime \right| ^2 + \xi _3^2 \right) }^{-1} \) are \(\xi _3 = \pm i s_{\lambda } \), the residue theorem implies the partial Fourier inverse transform of \( { \left( \lambda + \left| n^\prime \right| ^2 + \xi _3^2 \right) }^{-1} \) with respect to \(\xi _3\) is given by inserting \(\xi _3 = i s_{\lambda } \, \mathrm {or} \, - i s_{\lambda }\) into \( e ^{ i x_3 \xi _3}\) so that the real part become to be negative. Thus, we have
Moreover, this formula leads to
Combining the above two calculations and the formula
we obtain (25).
2.3 Boundedness of the anisotropic Helmholtz projection
Next, we consider the equation (III) with boundary data \(\phi = ( \phi _+, \phi _- ) \in C_0^\infty (\varOmega )\). Applying the partial Fourier transformation to (III), we have
for \(n^\prime \in \mathbb {Z}^2 {\setminus } \{ 0 \}\) and \(x_3 \in (-1 , 1)\). The solution to (27) is of the form
for some constant \(C_1\) and \(C_2\). Take the constants so that (27) satisfied, namely
then the solution to (27) is given by
Moreover, its anisotropic gradient given by
We apply the trace to (28) to get
We insert \(\phi _{\pm } = \gamma _{\pm } {\mathbb {P}}_{\epsilon } ^{ \mathbb {T} ^2 \times \mathbb {R} } f\) to (28) for \(f \in C_0^\infty (\varOmega )\) satisfying \(\mathrm {ave}_H (f)=0\) and set
Lemma 13
Let \(1< q < \infty \), \(0 < \epsilon \le 1\) and \(s \ge 0\). Then, there exists a constant \(C = C ( q ) \), which is independent of \(\epsilon \), the operator \(\varPi _\epsilon \) can be extended to a bounded operator from \(W^{s,q}_{\mathrm {af}} (\varOmega )\) into itself such that
for all \(f \in W^{s, q}_{\mathrm {af}} ( \varOmega ) \).
Proof
Let \(f \in C_0^\infty (\varOmega )\) satisfy \(\mathrm {ave}_H (f) = 0\). We seek the multiplier of \(\varPi _\epsilon \) by a direct calculation. Recall that the symbol of \( {\mathbb {P}}_{\epsilon } ^{ \mathbb {T} ^2 \times \mathbb {R} }\) is of the form
Since the symbol of \( {\mathbb {P}}_{\epsilon } ^{ \mathbb {T} ^2 \times \mathbb {R} } \) have poles at \(\xi _3 = \pm i \epsilon \left| {n}^{\prime } \right| \), we apply \(e_3 \cdot \) to (30) by the left-hand side and use the residue theorem so that the power of e is negative to get
Note that the integration is due to the relationship between the Fourier transform and convolution. Applying trace operators \(\gamma _{\pm }\) and \(\alpha _{\epsilon , \pm } ( {n}^{\prime } , x_3 ) \), respectively, and taking Fourier inverse transformation with respect to \( {n}^{\prime } \), we get
By the definition of \(\alpha _{\pm , \epsilon }\),
Symbols in the integral are written by \(A ( \epsilon {n}^{\prime } ) ( 1 + \epsilon \left| {n}^{\prime } \right| ) e ^{- \epsilon \left| {n}^{\prime } \right| ( \left| x_3 \pm 1\right| + \left| \zeta \pm 1\right| ) } \) with a symbol A. The Mikhlin constant of \(A ( \epsilon {n}^{\prime } ) \) is independent of \(\epsilon \) by Proposition 10. The same argument is valid for \(I_j\) (\(j = 2, 3, 4\)). Thus, we conclude by Propositions 8 and 9 that
for all \(f \in C_0^\infty (\varOmega )\) satisfying \(\mathrm {ave}_H (f) = 0\), where the constant C is independent of \(\epsilon \). Thus, the estimate (29) holds for \(s = 0\). We find from the formula (32) that \(\partial _j\) commutes with \(\varPi _\epsilon \) for \(j = 1, 2\). Moreover, the equation (27) implies
Thus, we conclude that (29) holds for all positive even numbers. We obtain (29) for all \(s>0\) by interpolation. \(\square \)
We set the operator
for all \(f \in C_0^\infty (\varOmega )\) satisfying \(\mathrm {ave}_H (f)=0\). Then, Lemma 13 implies
Corollary 14
Let \(1< q < \infty \), \(0 < \epsilon \le 1\) and \(s \ge 0\). Then, there exists a constant \(C > 0\), which is independent of \(\epsilon \), the operator \(P_{N, \epsilon }\) can be extended to a bounded operator from \(W^{s,q}_{\mathrm {af}} (\varOmega )\) into itself such that
for all \(f \in W^{s, q}_{\mathrm {af}} (\varOmega )\).
Remark 15
Note that \(P_{N, \epsilon }\) is not the anisotropic Helmholtz projection on \(\varOmega \). \(P_{N, \epsilon }\) is the operator which maps from the \(L^q\)-vector fields into \(L^q\)-divergence-free vector fields with tangential trace. However, we find that the anisotropic Helmholtz projection is bounded from \(L^q (\varOmega )\) into itself by the same method of Lemma 13.
Lemma 16
Let \(1< q < \infty \) and \(0 < \epsilon \le 1\). Then, there exists a constant \(C > 0\), which is independent of \(\epsilon \), such that
for all \(f \in L^q (\varOmega )\).
Proof
Let \(u \in C^\infty _0 (\varOmega )\). Then, we obtain the solution \(\pi _\epsilon \) to the Neumann problem
The anisotropic Helmholtz projection \(H_\epsilon \) is represented by
In the case of the Dirichlet boundary condition, i.e., \(\gamma _\pm u = 0\), the right-hand side of the second equality of (33) is zero. Let us consider the \(L^q\)-boundedness of \(\nabla _\epsilon \pi _\epsilon \), which implies the boundedness of the anisotropic Helmholtz projection. For the solution \(\pi ^0\) to the equation
we have
Let \(\pi _\epsilon ^1\) and \(\pi _\epsilon ^2\) be the solutions to
and
respectively, for \(u \in C^\infty _0 (\varOmega )\) satisfying \(\mathrm {ave}_H (u)=0\).
We first consider (35). It follows from integration by parts that
This formula, the Mikhlin theorem and Proposition 5 imply
where \(C>0\) is independent of \(\epsilon \). Moreover, since \(e_3 \cdot \nabla _\epsilon \varDelta _\epsilon ^{-1} \mathrm {div}_\epsilon \) is given by the left-hand side of (31), we use the same method as in Lemma 13 to get
where \(C>0\) is also independent of \(\epsilon \). The formula (34) and estimates (36) and (37) imply \(L^p\)-boundedness of the anisotropic Helmholtz projection on \(\varOmega \). \(\square \)
Proposition 17
Let \(0 <\epsilon \le 1\), \( 1< q < \infty \), \(0< a < 1/2\), \(z \in \mathbb {C} \) satisfying \(- a< \mathrm {Re} z < 0\) and \( 0< \theta < \pi / 2\). Then, there exists a constant \(C = C ( q, a, \theta ) \), which is independent of \(\epsilon \), the solution \( \pi _3 \) to (III) with boundary data \( ( \gamma K_{\lambda , \epsilon } f \cdot \nu ) \nu \) satisfies
for all \(f \in L^{ q } ( \varOmega ) \).
Proof
In view of Remark 7, we may assume \(\mathrm {ave}_H (f)=0\) without loss of generality. Since
and the Cauchy integral commutes with \( {\varPi }_{\epsilon } \), the conclusion is obtained from Proposition 11 and Lemma 13. \(\square \)
Proposition 18
Let \(0 <\epsilon \le 1\), \( 1< q < \infty \), \( 0< \theta < \pi / 2\) and \(\lambda \in \varSigma _\theta \). Then, there exists a constant \(C = C ( q, \theta ) \), which is independent of \(\epsilon \), the solution \( \pi _3 \) to (III) with boundary data \( ( \gamma K_{\lambda , \epsilon } f \cdot \nu ) \nu \) satisfies
for all \(f \in L^{ q } ( \varOmega ) \).
Proof
The estimate (39) is a direct consequence of (24), (38), Lemma 13 and Proposition 12. \(\square \)
2.4 Estimate for \(v_2\)
Let us consider the equation (II) with tangential boundary data \(g = ( g_{+}, g_{-} ) \). Set
Then, \(y_{ \lambda , \epsilon }\) satisfies
where \(k^\prime _{\lambda , \epsilon }\) is defined by (25). We define the multiplier operator \(L_{\lambda , \epsilon }\) as
where \(e^\prime _\lambda \) is defined by (26). Let \(p_{\epsilon }^{\prime } ( {n}^{\prime } , x_3 ) \) be a partial Fourier transform of the symbol of \( {\mathbb {P}}_{\epsilon } ^{ \mathbb {T} ^2 \times \mathbb {R} }\) with respect to \(\xi _3\). Then, we obtain
where \( \cdot *_3 \cdot \) is convolution with respect to \(x_3\). We set
Then, \(W_{\lambda , \epsilon } g\) is a solution to (II) with boundary data \( \gamma W_{\lambda , \epsilon } g\). We first get the Fourier multiplier of \( \gamma W_{\lambda , \epsilon }\). Next, we show the map \(S_{\lambda ,\epsilon } : g \mapsto \gamma W_{\lambda , \epsilon } g\) has a bounded inverse for large \(\lambda \). Put
then, \(V_{\lambda , \epsilon } g\) gives the solution to (II) with boundary data g.
Proposition 19
Let \( 1< q < \infty \), \(0 < \epsilon \le 1\), \(s\ge 0\) and \(0< \theta < \pi /2 \). Then, there exist \(r > 0\) and, for \(\lambda \in \varSigma _{\theta }\) satisfying \(\left| \lambda \right| > r \), a bounded operator \(R_{\lambda , \epsilon }\) from \(W^{s,q} ( \mathbb {T} ^2)\) into itself satisfying
and
where \(C>0\) is independent of \(\epsilon \), such that
Proof
Let \(g \in C^\infty ( \mathbb {T} ^2)\) be horizontal average-free. Since \(e^\prime _{\lambda }\) is an even function with respect to \(x_3\), we find from the change of variable that
and similarly
Thus, we find from (43) that
We apply \(P_{N, \epsilon }\) to (49) to get
Let us estimate \(I_1\) and \(I_3\). The identity (41) implies
We show the other terms are \(O(1 / \left| \lambda \right| ^{1/2})\). By (25) and (40), we have
We find from (16) and (17) in Proposition 10 and the estimate
that
where we interpret that the multiplier \(II_1\) is extended from \(\mathbb {Z}^2\) to \( \mathbb {R} ^2\) by the canonical way. Since
by the same way as above, we have
for \(\lambda \in \varSigma _\theta \), where the constants c and \(C > 0\) are independent of \(\epsilon \). Note that \(II_3\) has a little bit problem near \(\epsilon = 0\) since we cannot use the decay of \( e ^{- 2 \epsilon \left| {\xi }^{\prime } \right| }\) to obtain uniform boundedness of the Mikhlin constant at this point. However, we can use the decay of \(1/ ( \lambda + (1 - \epsilon ^2) \left| {\xi }^{\prime } \right| ^2 ) \) around \(\epsilon = 0\). On the other hand, when \(\epsilon \) is away from 0, we have no problem to use decay of \(e^{- 2 \epsilon \left| \xi ^\prime \right| }\). Thus, combining this observation with Proposition 9, (52) and (54), we conclude that
where \(C>0\) is independent of \(\epsilon \). Thus, we find from (53), (55) and (56) that
Next, we estimate \(I_2\) and \(I_4\). It follows from (25) that
Recall \( \partial _3 \eta _{\lambda , \epsilon } ( {n}^{\prime } , \pm 2) = \frac{ \epsilon ^2 }{ \lambda + ( 1 - \epsilon ^2 ) \left| n^{\prime } \right| ^2 } \frac{ e ^{- 2 s_{\lambda }} - e ^{ - 2 \epsilon \left| n^{\prime }\right| } }{ 2 } \). Then, we find from the first inequality of (16) and (54) that
and
where \(C>0\) is independent of \(\epsilon \). The formula (28), estimates (17) and (18) lead to
We conclude from Proposition 5 that
where C is independent of \(\epsilon \). Thus, taking \(\left| \lambda \right| \) sufficiently large; clearly, the choice of \(\lambda \) is also independent of \(\epsilon \); we conclude by (50), (51), (57) and (60) that
By the Neumann series argument, we obtain (46) for \(s = 0\). Moreover, we find from (53), (55), (56) and (59) that (47) holds for \(s = 0\). Since \(\partial _j\) (\(j = 1,2\)) commutes Fourier multiplier operators, we obtain (46) and (47) for \(s > 0\). \(\square \)
Proposition 20
Let \(1< q < \infty \), \(0 < \epsilon \le 1\), \(0< \theta < \pi / 2\) and \(\lambda \in \varSigma _\theta \). Then, there exist \(r > 0\) and a constant \(C > 0\), which is independent of \(\epsilon \) and \(\lambda \), if \(\left| \lambda \right| \ge r\), \(V_{\lambda , \epsilon }\) defined by (45) satisfies
for all \(g \in L^{ q } ( \partial \varOmega ) \) satisfying \(\mathrm {ave}_H (g)=0\).
Proof
We take \(r>0\) so that \(R_{\lambda , \epsilon }\) exists. Then, \(S_{\lambda , \epsilon }^{-1}\) is bounded on \(L^q (\varOmega )\). We find from the resolvent estimate for the Dirichlet Laplacian on \(\varOmega \), see Lemma 5.3 in [1] and Proposition 19 that
where \(C>0\) is independent of \(\epsilon \). By Corollary 14, we obtain (61). \(\square \)
Proposition 21
Let \(1< q < \infty \) and \(0 < \epsilon \le 1\). Then, there exists a constant \(C > 0\), which is independent of \(\epsilon \), such that
for all \(f \in L^{ q } _{\mathrm {af}} (\varOmega )\).
Proof
Since the symbol has poles at \(\xi _3 = \pm i \epsilon \left| {n}^{\prime } \right| \), we calculate its partial Fourier transform with respect to \(\xi _3\) by the residue theorem and obtain
This formula and Proposition 5 imply
where C is independent of \(\epsilon \). Combining this estimate with the boundedness of \(\mathbb {P}_\epsilon ^{ \mathbb {T} ^2 \times \mathbb {R} }\), we obtain (62). \(\square \)
Let us show BIP for the solution operator for the equation (II).
Proposition 22
Let \(1< q < \infty \), \(0 < \epsilon \le 1\), \(0< \theta < \pi /2\), \(\lambda \in \varSigma _\theta \), \(0< a < 1/2\), and z satisfying \( - a< \mathrm {Re} z <0 \). Then, there exists a constant \(C = C ( q, a, \theta ) \), it holds that
for all \(f \in L^{ q } (\varOmega )\), where \( v_1 = K_{\lambda , \epsilon } f\).
Proof
In view of Remark 7, we may assume \(\mathrm {ave}_H (f)=0\) without loss of generality. It holds by (24) that
We find from this formula, (28), (42), (44), (45) and (48) that the integrand of the left-hand side of (63) can be essentially written as
where ± should be take properly.
It follows from (16) and (52) that
Let \(R>0\) be large enough so that \(S_{\lambda , \epsilon }^{-1}\) in Proposition 19 exists. Then, we find from the change of integral curve around the origin to ensure \(\left| \lambda \right| > R\) and Proposition 5 that
for some \(a , b \in \{ -1 , 1 \}\), where C and \(C_R\) are independent of \(\epsilon \). Applying Proposition 8, we obtain
It follows from (26), (28), (40) and Proposition 10 that
Thus, we find from Proposition 21 that
By Proposition 19, the trace theorem, Lemma 13 and the resolvent estimate for the Laplace operator on \( \mathbb {T} ^2 \times \mathbb {R} \), we have
for some small \(\delta >0\). The resolvent estimate for the Dirichlet Laplacian on \(\varOmega \), see Lemma 5.3 in [1], and Lemma 13 imply
for some small \(\delta >0\). We find from the above two inequalities
Thus, we find from the change of integral line around the origin that
where \(C>0\) is independent of \(\epsilon \). \(\square \)
Proof of Lemma 6
Lemma 6 is a direct consequence of Propositions 11, 17 and 22.
\(\square \)
We next prove Lemma 3 from Lemma 6. For this purpose we need further uniform estimate for the resolvent to compare \(\Vert \nabla ^2 u \Vert _{L^q (\varOmega )}\) and \(\Vert A_\epsilon u \Vert _{L^q (\varOmega )}\). For resolvent estimates we begin with
Proposition 23
Let \(1< q < \infty \), \(0 < \epsilon \le 1\), \(0< \theta < \pi /2\). Let \(\lambda \in \varSigma _\theta \) be sufficiently large so that \(S_{\lambda , \epsilon }^{-1}\) exists in Proposition 19. Then, there exists a constant \(C = C ( q, \theta ) \), it holds that
for all \(f \in L^{ q } (\varOmega )\), where \( v_1 = K_{\lambda , \epsilon } f\).
Proof
In view of Remark 7, we may assume \(\mathrm {ave}_H (f)=0\) without loss of generality. It is enough to estimate the second derivative of the left-hand side of (64) in \(L^q (\varOmega )\). We find from (16) and (65) that
where \(C> 0\) is independent of \(\epsilon \). Similarly, it follows from (40), (28) and Proposition 10 that
Thus, we find from Corollary 14 and Proposition 8 that
where \(\nabla _H = (\partial _1, \partial _2)^T\), \(I_j\) is defined in (64) and \(C>0\) is independent of \(\epsilon \). Since
we use the same way as above to get
where \(I_j\) is defined in (64) and \(C>0\) is independent of \(\epsilon \). Propositions 12 and 19, Lemma 13 and the trace theorem imply
and its norm is bounded uniformly on \(\epsilon \). By the definition of the operator \(L_{\lambda , \epsilon }\), see (42), we have
solves the elliptic equations \(\lambda u - \varDelta u =0\). Moreover, the boundary data belong to \(W^{3 - 1 / q, q} ( \mathbb {T} ^2)\) by (26), (40) and Proposition 5. Thus, we find from (45), Corollary 14 and smoothing effect of the solution operator to the elliptic equation that
where \(\delta >0\) is small and C is independent of \(\epsilon \). \(\square \)
Lemma 24
Let \(1< q < \infty \), \(0 < \epsilon \le 1\), \(0< \theta < \pi / 2\) and \(\lambda \in \varSigma _\theta \) satisfying \(\left| \lambda \right| > R\) for sufficiently large \(R>0\). Then, there exists a constant \(C = C ( q, \theta ) \) such that
for all \(f \in L^{ q } (\varOmega )\).
Proof
This is a direct consequence of Propositions 12, 18 and 23. \(\square \)
Lemma 25
Let \(1< q < \infty \), \(0 < \epsilon \le 1\). Then, there exists a constant \(C = C ( q ) \) such that
for all \(u \in D(A_\epsilon )\).
Proof of Lemma 3
Let u be a solution of (4). Our uniform BIP yields
by the Dore–Venni theory, where \(C>0\) is independent of \(\epsilon \) and T. Applying an a priori estimate Lemma 25, we can replace \(\Vert A_\epsilon u \Vert _{\mathbb {E}_0 (T)}\) by \(\Vert \nabla ^2 u \Vert _{\mathbb {E}_0 (T)}\). Since \((u, \pi )\) solves (4) and \(\partial _t u\) and \(\nabla ^2 u\) are controlled, we are able to estimate \(\Vert \nabla _\epsilon \pi \Vert _{\mathbb {E}_0 (T)}\). This completes the proof of Lemma 3. \(\square \)
It remains to prove Lemma 25. We first observe an a priori estimate slightly weaker than Lemma 25, which is proved by using the resolvent estimate Lemma 24.
Proposition 26
Let \(1< q < \infty \) and \(0 < \epsilon \le 1\). There exists a unique solution \((u, \pi ) \in D (A_\epsilon ) \times L^q (\varOmega ) / \mathbb {R} \) to
for \(f \in L^q (\varOmega )\), such that
where \(C > 0\) is independent of \(\epsilon \) and f.
Proof
The equations are equivalent to
for sufficiently large \(\lambda _0 > 0\). We find from Lemma 24 that
for some constant \(C> 0\), which is independent of \(\epsilon \). The first equation in (69) implies
For uniqueness we multiply the first equation by u and integrating by parts yields \(\nabla u = 0\). By the Poincaré inequality, it implies \(u = 0\). This argument works for \(q \ge 2\) since \(\varOmega \) is bounded. Since \((\lambda _0 + A_\epsilon )^{-1}\) is compact in \(L^q (\varOmega )\), the Riesz–Schauder theorem implies that 0 is in resolvent since \(\mathrm {ker} A_\epsilon = \{ 0 \}\). In particular, (69) is uniquely solvable for any \(f \in L^q (\varOmega )\) for \(q \ge 2\). By duality argument the solvability of \(q \ge 2\) implies the uniqueness of (69) for \(1< q < 2\). Again by compactness of \((\lambda _0 + A_\epsilon )^{-1}\) the solvability for (69) follows. \(\square \)
Proof of Lemma 25
Assume that if the statement were false, then there would exist a sequence \(\{ \epsilon _k \}_{k\in \mathbb {Z}_{\ge 1}}\), (\(0 < \epsilon _k \le 1\)) and \(u_k \in D(A_\epsilon )\) such that
Since the problem is linear we may assume that
By \(A_{\epsilon _k} u_k = f_k\) and Proposition 26, we have
for some constant \({C}>0\), which is independent of \(\epsilon _k\). Letting \(k \rightarrow \infty \) implies
By the Poincaré inequality for \(u_k\), our bound \(\Vert \nabla u_k \Vert _{L^q (\varOmega )}\) implies that \(u_k\) and \(\nabla u_k\) are bounded in \(L^q (\varOmega )\). By Rellich’s compactness theorem, we observe that \(u_k \rightarrow u\) for some \(u \in L^q (\varOmega )\) strongly in \(L^q (\varOmega )\) by taking a subsequence. The estimate (70) implies that
We may assume \(\epsilon _k \rightarrow \epsilon _*\in [0, 1]\) and \(u_k \rightarrow u\) as \(k \rightarrow \infty \) by taking a subsequence. The situation is divided into two cases, i.e., \(\epsilon _*= 0\) or \(\epsilon _*> 0\). By definition,
with some function \(\pi _k\) satisfying \(\int _{\varOmega } \pi _k \mathrm{d}x = 0\). Since \(\Vert \nabla ^2 u_k \Vert _{L^q (\varOmega )} \le 1\), we see that
By the Poincaré inequality, \(\{ \pi _k \}\) is bounded in \(L^q (\varOmega )\). By Rellich’s compactness theorem we may assume \(\pi _k \rightarrow \pi \) in \(L^q (\varOmega )\) for some \(\pi \in L^q (\varOmega )\) strongly by taking a subsequence. If \(\epsilon _*= 0\), this implies \(\pi \) is independent of z. Since \(\mathrm {div}_{\epsilon _k} \, u_k = 0\) and the vertical component \(w_k = 0\) on \(x_3 = \pm 1\), integration vertically on \((-1 , 1)\) yields that the horizontal limit v satisfies
where \(\mathrm {div}_H = \nabla _H \cdot \). Thus, the horizontal component v satisfies the hydrostatic Stokes equations
Since we know the only possible \(W^{2 , q}\)-solution is zero, so we conclude that \(v = 0\). Since \(\Vert \nabla v_k \Vert _{L^q (\varOmega )}\) is bounded, \(\mathrm {div}_{\epsilon _k} \)-free condition implies that the horizontal limit w is independent of the vertical variable. By the boundary condition \(w = 0\) at \(x_3 = \pm 1\), this implies w must be zero. We thus observe that \(u_k \rightarrow 0\) strongly in \(L^q (\varOmega )\), this contradicts \(\Vert u \Vert _{L^q (\varOmega )} \ge 1 / {C} > 0\). The case \(\epsilon _*\) is easier since the limit satisfies the anisotropic Stokes equations
By the uniqueness \(u \equiv 0\) in \(\varOmega \). This again contradicts \(\Vert u \Vert _{L^q (\varOmega )} \ge 1 / {C} > 0\). The proof of Lemma 25 is now complete. \(\square \)
As an application of Lemma 3, we obtain
Corollary 27
Let \(p,q\in (1,\infty )\), \(T>0\), \(F = (f_H, f_z) \in \mathbb {E} _0(T)\), \(U_0\in X_\gamma \) and \(0<\epsilon \le 1\). Then, there is a unique solution \(( U_\epsilon , P_\epsilon ) \in \mathbb {E}_1 (T) \times \mathbb {E}_0(T)\) to the equations
where P is unique up to a constant. Moreover, there exist constants \(C>0\) and \(C_T>0\), which is independent of \(\epsilon \), such that
Proof
Lemma 3 implies there exists a solution \((\tilde{U}, \tilde{P})\) to (4) with initial data \(U_0\) such that
Set
Then, (U, P) is the desired solution satisfying (72). Note that the spectrum of the anisotropic Stokes operator is positive. This implies exponential decay of \(\Vert (V, \epsilon W) \Vert _{{B^{2 (1 - 1 / p)}_{q, p} (\varOmega )}}\). Thus, the constant \(C_T\) is uniformly bounded on T. \(\square \)
3 Nonlinear estimates and regularity for w
In this section, we begin by recalling some estimates of products of functions. Then, we estimate terms \(F_H\), \(F_z\), F and derive necessary regularity of w. Although the following propositions have been already proved in [11], we restate them to explain our restriction for p and q and for the reader’s convenience.
Proposition 28
Lemma 4. 3 in [11] Let \(T > 0\), \(p, q \in ( 1, \infty ) \) such that \(2 / 3p + 1 / q \le 1\). Then, there exists a constant \(C = C (p, q) > 0\) such that
for all \(v_1, v_2 \in \mathbb {E}_1(T)\).
Proposition 29
Lemma 4. 5 in [11] Let \(T > 0\) and \(x_3 \in ( - 1, 1 ) \). Let \(p, q \in ( 1, \infty ) \) such that \( 1 / p + 1 / q \le 1 \). Then, there exists a constant \(C = C (p, q) > 0\) such that
for all \(v_1, v_2 \in \mathbb {E}_1 (T)\) and \( w_1 : = - \int _{-1}^{x_3} \mathrm {div}_H v_1 \ \mathrm{d} \zeta \).
Proposition 30
Let \(T > 0\) and \(x_3 \in ( - 1, 1 ) \). Let \(p, q \in ( 1, \infty ) \) such that
Then, there exists a constant \(C = C (p, q) > 0\) such that
for all \(v_1 , v_2 \in \mathbb {E}_1 (T)\), \(-1 \le x_3 \le 1\), \(j = 1, 2\) and \(k = 1, 2, 3\).
Proof
We find from the Hölder inequality that
Applying \(L^p\)-norm for the time variable and using the Hölder inequality again, we have
By the Sobolev inequality and the mixed-derivative theorem (interpolation inequality)
we have
for \(i = 1,2\) and p, q satisfying \(1/ 2p + 1 / q \le 3 / 4\) and \(q \le 2\). For the mixed derivative theorem, the reader is referred to the book [32]. In the case \(q \ge 2\), we find from the Sobolev inequality and the mixed-derivative theorem that
for \(i = 1,2\) and p, q satisfying \(1/ p + 1 / q \le 1\). By the Sobolev inequality and the mixed-derivative theorem, we get
for p, q satisfying \(1/ 2p + 1 / q \le 3 / 4\) and \(q \le 2\). We next consider the case \(q \ge 2\). We argue in same way as above to get
for p, q satisfying \(1/ p + 1 / q \le 1\). Thus, we obtain (73). \(\square \)
Proposition 31
Let \(T > 0\), \(1< p,q < \infty \) and \(s > 1/q\). Let \(w_0 \in {B^{2 (1 - 1 / p)}_{q, p} (\varOmega )}\) and \(v \in \mathbb {E}_1(T)\). Let f be a quadratic nonlinear function and g be a bi-linear function satisfying
for \(v_1 \in W^{1, p} (0, T ;W^{s , q} (\varOmega ) ) \cap L^{p} (0, T ;W^{2 + s, q} (\varOmega ) )\) and \(v_2 \in \mathbb {E}_1 (T)\). Then, there is a constant \(\delta > 0\) depending only on p, q, v and f such that if T is taken so that \(\Vert v \Vert _{\mathbb {E}_1 (T)} \le \delta \), the solution w to
exists and satisfies the estimate
with some constant \(C > 0\).
Proof
Let \(u = \mathcal {S} (h, w_0)\) solve
for an external force \(h \in \mathbb {E}_0 (T)\) and initial data \(w_0 \in {B^{2 (1 - 1 / p)}_{q, p} (\varOmega )} \). The operator \(\mathcal {S}\) has the maximal regularity of the form
We define the linear bounded operator \(H : \mathbb {E}_0 (T) \rightarrow \mathbb {E}_0 (T)\) by
for \(h \in \mathbb {E}_0 (T)\). By the definition of g and (75), we get
If we take T sufficiently small so that
we have
Using the Neumann series argument, we find that for small T the inverse operator of H exists such that
Put
By the definition and (76), we obtain
We now check that w is the solution to (74). It follows from the definition of w that
\(\square \)
Let us show \(w \in \mathbb {E}_1 ( T ) \). In our previous paper [11], we first derive the equation which w satisfies by applying \(\int _{-1}^{x_3} \mathrm {div}_H \, \cdot \ \mathrm{d} \zeta \) to the equations v satisfies. Then, estimating the corresponding nonlinear terms and applying the maximal regularity principle, we obtain \(w \in \mathbb {E}_1 ( T ) \). Note that, in the present paper, we invoke additional regularity for v to deal with the trace of the second derivative.
Although, in [15], the authors treat higher-order regularity of the solution to the primitive equations, they do not explicitly write the maximal regularity in fractional Sobolev spaces. However, it is easy to modify their proof to get the maximal regularity in the fractional Sobolev spaces. In [16], the argument to get \(H^\infty \)-calculus of hydrostatic Stokes operator is based on \(H^\infty \)-calculus for the Laplace operator and perturbations arguments. Since the Laplace operator admits \(H^\infty \)-calculus in fractional Sobolev spaces, it is not difficult to establish \(H^{\infty }\)-calculus of the hydrostatic Stokes operator in fractional Sobolev spaces. We also find local well-posedness of the primitive equations in fractional maximal regularity space \(W^{1, p} (0, T ;W^{s , q} (\varOmega ) ) \cap L^{p} (0, T ;W^{2 + s, q} (\varOmega ) )\) for \(s > 1 / q\) in the same way [15] to get local well-posedness, namely, using Lemma 6.1, Corollary 6.2 and Theorem 5.1 in [15].
Remark 32
It is already known that \(v \in \mathbb {E}_1 (T)\) for initial data \(v_0 \in X_{\gamma }\) by Giga et al. [15, 16].
Proof of Lemma 4
Integrating (PE) both sides over \( ( -1, 1 ) \), we find \( ( \overline{v}, \overline{\pi } ) \) satisfy
It is clear that \(\tilde{v} := v - \overline{v}\) satisfies
Then, \(\tilde{u} = (\tilde{v}, w)\) solves
Note that the pressure term no longer appears in the above equations and
Applying \( - \mathrm {div}_H \) to (78) and integrating over \((-1, x_3)\) with respect to vertical variable, we find
with initial data \(w_0\). Since \(v_0 \in B^{s}_{q,p}(\varOmega )\) for \(s > 2 - 2/ p + 1/q\), we have \(v \in \mathbb {E}_1 (T) \cap L^p (0, T ; W^{2 + 1/q +\delta , q}(\varOmega ))\) for some \(\delta >0\) by [15] and [16], and thus, \(\left| \left| I_1\right| \right| {\mathbb {E}_0 ( T ) } \le C\) for some \(C>0\). We use integration by parts to get
where \(j = 1, 2\) and Einstein’s summation convention is used. We find from Propositions 28, 29 and 30 that
and
Similarly, \(I_3\) is decomposed into w-depend part \(I_{31} (v, w)\) and w-independent part \(I_{32} (v)\) and estimated as
and
Thus, we find from Proposition 31 that
for some constant \(C > 0\) and small \(T^\prime \). The solution to the primitive equations v is smooth in the time interval \([T^\prime , T)\). The formula \(w = \int _{-1}^{x_3} \mathrm {div}_H v \, \mathrm{d} \zeta \) implies that w is also smooth in on \([T^\prime , T)\). Thus, we conclude
\(\square \)
4 Justification of the hydrostatic approximation and global-well-posedness of the anisotropic Navier–Stokes equations
Let us prove our main theorem. Recall that \(u = (v, w)\) is the solution to the primitive equations and \(U_\epsilon = (V_\epsilon , W_\epsilon )\) is the solution to (1). We construct the solution \(u_\epsilon = (v_\epsilon , w_\epsilon )\) to (SNS) of the form \(u_\epsilon = u + U_\epsilon \). The key is construction of \(U_\epsilon \) by iteration. Note that our idea to construct the solution is based on the principle which small data implies the global well-posedness.
Proof of Theorem 1
Let \(C_1\) be the maximum of constants C in Propositions 28 and 29, (72) and the constant in the trace theorem. Let us construct a solution \((V_{\epsilon }, {\epsilon } W_{\epsilon })\) to (1) with zero initial data on \([0, {{{\mathcal {T}}}}]\). Set \((u_{\epsilon }, p_{\epsilon }) := (v + V_{\epsilon }, w + W_{\epsilon }, p + P_{\epsilon })\), then this is the desired solution to (SNS). We denote by \(\Vert \cdot \Vert _{\mathbb {E}_1 (a, b)}\) and \(\Vert \cdot \Vert _{\mathbb {E}_0 (a, b)}\) the \(\mathbb {E}_1\)-norm and \(\mathbb {E}_0\)-norm on the time interval [a, b), respectively. We choose \(0 < { {\mathcal {T}}} \le 1\) so small that
for sufficiently large integer N and
for all integer \(m \in [1, N]\). The choice of \({ {\mathcal {T}}}\) depends on T and u but is independent of \(\epsilon \). We divide the time interval [0, T] into \(\cup _{m = 0}^N [m { {\mathcal {T}}}, (m + 1) { {\mathcal {T}}} ]\). We denote the left-hand side of (1) by
Clearly, the choice of \({\mathcal {T}}\) is independent of F. We denote the solution (U, P) to (71) with initial data \(U_0\) and external force F by
We inductively set
Propositions 28, 29 and Corollary 27 lead to
This quadratic inequality and (84) imply
for \({C^*= \left( 1/10 + 1/ 100 C_1 \right) }\) and small \(\epsilon > 0\). We set the differences
Then, seeking the equation which \((\widetilde{U}_{\epsilon , j}, \widetilde{P}_{\epsilon , j})\) satisfies and applying Propositions 28, 29 and Corollary 27, we have
Thus, \((U_{\epsilon }, P_{\epsilon }) : = ( \lim _{j \rightarrow \infty } U_j, \lim _{j \rightarrow \infty } P_j ) = (\sum _{j = 0} \tilde{U}_{\epsilon , j}, \sum _{j = 0} \tilde{P}_{\epsilon , j})\) exists in \(\mathbb {E}_1 (\mathcal {T})\) and satisfies
By construction \((U_{\epsilon }, P_{\epsilon })\) satisfies (1) on \([0, {\mathcal {T}}]\). Moreover, by trace theorem there exists a constant \(C_{tr} > 0\) such that
We next construct the solution to (1) on \([ {\mathcal {T}}, 2{\mathcal {T}}]\) with initial data
We set
Then, \((a_{\epsilon , 1}, \pi _{\epsilon , 1})\) solves
Corollary 27 implies
Let the vector field \(a_{\epsilon } = (b_{\epsilon }, c_{\epsilon })\) be the solution to
If we put \(U_{\epsilon } := a_{{\epsilon }, 1} + a_{\epsilon }\) and \(P_{\epsilon } := \pi _{\epsilon , 1} + \pi _{\epsilon }\), then \((U_\epsilon , \varPi _\epsilon )\) is the solution to (1) with initial data \(U_{\epsilon } (\mathcal {T})\). We inductively set
for \(j \ge 1\). Applying Propositions 28, 29 and Corollary 27 to (91), we find
If we take \({\epsilon }\) so small that
we have
Thus, we inductively obtain
for all \(j \ge 1\). Set
Applying Propositions 28, 29 and Corollary 27 to the equations that
satisfies, we find
The last inequality holds if \({\epsilon }\) is sufficiently small. Thus,
exists in \(\mathbb {E}_1 ({{\mathcal {T}}}, 2 {{\mathcal {T}}})\) and satisfies (91) such that
The functions \((U_{\epsilon }, P_{\epsilon })\) solves (1) on the time interval \([{{\mathcal {T}}}, 2 {{\mathcal {T}}}]\) with initial data \(U_{\epsilon } ({{\mathcal {T}}})\) such that
where we used (90) in the last inequality. By induction, the solution \((U_{\epsilon }, P_{\epsilon })\) constructed by the same way on the time interval \([m {{\mathcal {T}}} , (m + 1) {{\mathcal {T}}}]\) inductively satisfies
where \(\beta _m\) is defined by \(\beta _0 = 1 + 2 C_{tr} C_{\mathcal T}\) and \(\beta _m = 1 + 2 C_{tr} C_{\mathcal T} \beta _{m-1}\) for \(m = 1, 2, \dots , N - 1\). Since T is finite, the induction ends in finite steps. Thus, we conclude
In the case of \(T = \infty \), we first show (96) for a sufficiently large \(T^\prime > 0\). Since \(\Vert u \Vert _{\mathbb {E}_1 (T^\prime , \infty )}\) is small, we extend the existence time of \((U_\epsilon , P_\epsilon )\) from \([0, T^\prime )\) to \([0, \infty )\) by one step. \(\square \)
Remark 33
It is worth pointing out an upper bound of \(\epsilon \). We assume the estimate (95) holds for some m. In the proof of Theorem 1, we chose \(\epsilon \) so that the estimates (85), (87), (92), (93) and (94) hold. In order to extend the existence time of the solution \((U_\epsilon , \varPi _\epsilon ,)\) from \([0, (m + 1) \mathcal {T})\) to \([0, (m + 2) \mathcal {T})\), it suffices to choose \(\epsilon \) such that
Since \({\beta }_m\) is increasing in m, we have an example of upper bound of \(\epsilon \) as
where N is the integer in (83). The contribution of the initial data \(v_0\) in Theorem 1 is implicitly included in the choice of \(\mathcal {T}\).
Change history
26 May 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00028-022-00812-8
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The authors are grateful to Professor Matthias Hieber and Professor Amru Hussein for helpful discussions and comments. The authors are grateful to the anonymous referee for her/his careful reading and valuable comments.
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Dedicated to Professor Matthias Hieber on the occasion of his 60th birthday.
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K. Furukawa: The first author was partly supported by the Program for Leading Graduate Schools, Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, Japan Society for the Promotion of Science (JSPS). Y. Giga: The second author was partly supported by JSPS Grant-in-Aid for Scientific Research (Kiban) S (No. 26220702), A (No. 17H01091), A (No. 19H00639), B (No. 16H03948) and Challenging Pioneering Research (Kaitaku) (No. 18H05323). T. Kashiwabara: The third author was partly supported by JSPS Grant-in-Aid for Young Scientists B (No. 17K14230) and by Grant for The University of Tokyo Excellent Young Researchers. This work was partly supported by the DFG International Research Training Group IRTG 1529 and the JSPS Japanese- German Graduate Externship on Mathematical Fluid Dynamics.
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Furukawa, K., Giga, Y. & Kashiwabara, T. The hydrostatic approximation for the primitive equations by the scaled Navier–Stokes equations under the no-slip boundary condition. J. Evol. Equ. 21, 3331–3373 (2021). https://doi.org/10.1007/s00028-021-00674-6
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DOI: https://doi.org/10.1007/s00028-021-00674-6