Abstract
We consider the Navier–Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier’s slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global existence of a strong solution for large data. We also show several estimates for the strong solution with constants explicitly depending on the thickness of the thin domain. The proofs of these results are based on a standard energy method and a good product estimate for the convection and viscous terms following from a detailed study of average operators in the thin direction. We use the average operators to decompose a three-dimensional vector field on the thin domain into the almost two-dimensional average part and the residual part, and derive good estimates for them which play an important role in the proof of the product estimate.
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Acknowledgements
This work is an expanded version of a part of the doctoral thesis of the author [33] completed under the supervision of Professor Yoshikazu Giga at the University of Tokyo. The author is grateful to him for his valuable comments on this work and to Mr. Yuuki Shimizu for fruitful discussions on Killing vector fields on surfaces. The author also would like to thank an anonymous referee for valuable remarks.
The work of the author was supported by Grant-in-Aid for JSPS Fellows No. 16J02664 and No. 19J00693, and by the Program for Leading Graduate Schools, MEXT, Japan.
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Appendices
Appendix A: Notations on Vectors and Matrices
In this appendix we fix notations on vectors and matrices. For \(m\in {\mathbb {N}}\) we consider a vector \(a\in {\mathbb {R}}^m\) as a column vector
and denote the i-th component of a by \(a_i\) or sometimes by \(a^i\) or \([a]_i\) for \(i=1,\dots ,m\). A matrix \(A\in {\mathbb {R}}^{l\times m}\) with \(l,m\in {\mathbb {N}}\) is expressed as
and the (i, j)-entry of A is denoted by \(A_{ij}\) or sometimes by \([A]_{ij}\) for \(i=1,\dots ,l\) and \(j=1,\dots m\). We denote the transpose of A by \(A^T\) and, when \(l=m\), the symmetric part of A by \(A_S:=(A+A^T)/2\). Also, we write \(I_m\) for the \(m\times m\) identity matrix. The tensor product of \(a\in {\mathbb {R}}^l\) and \(b\in {\mathbb {R}}^m\) is defined as
For three-dimensional vector fields \(u=(u_1,u_2,u_3)^T\) and \(\varphi \) on an open set in \({\mathbb {R}}^3\) let
Also, we define the inner product of \(3\times 3\) matrices A and B and the norm of A by
where \(\{E_1,E_2,E_3\}\) is an orthonormal basis of \({\mathbb {R}}^3\). Note that A : B does not depend on a choice of \(\{E_1,E_2,E_3\}\). In particular, taking the standard basis of \({\mathbb {R}}^3\) we get
for \(A,B,C\in {\mathbb {R}}^{3\times 3}\). Also, for \(a,b\in {\mathbb {R}}^3\) we have \(|a\otimes b|=|a||b|\).
Appendix B: Proofs of Auxiliary Lemmas
The purpose of this appendix is to present the proofs of Lemmas 4.1, 4.3, and 7.4. First we prove Lemma 4.1 after giving two auxiliary statements. Recall that \(\Gamma \) is a two-dimensional closed surface in \({\mathbb {R}}^3\) of class \(C^5\).
Lemma B.1
[34, Lemma B.4] Let U be an open set in \({\mathbb {R}}^2\), \(\mu :U\rightarrow \Gamma \) a \(C^5\) local parametrization of \(\Gamma \), and \({\mathcal {K}}\) a compact subset of U. For \(p\in [1,\infty ]\) if \(\eta \in L^p(\Gamma )\) is supported in \(\mu ({\mathcal {K}})\), then \(\eta ^\flat :=\eta \circ \mu \in L^p(U)\) and
If in addition \(\eta \in W^{1,p}(\Gamma )\), then \(\eta ^\flat \in W^{1,p}(U)\) and
where \(\nabla _s\eta ^\flat =(\partial _{s_1}\eta ^\flat ,\partial _{s_2}\eta ^\flat )^T\) is the gradient of \(\eta ^\flat \) in \(s\in {\mathbb {R}}^2\).
We omit the proof of Lemma B.1 since it is given in our first paper [34].
Lemma B.2
Let U be an open set in \({\mathbb {R}}^2\). Then
for all \(\varphi \in H_0^1(U)\).
The inequality (B.3) is the well-known Ladyzhenskaya inequality on \({\mathbb {R}}^2\) (see [25, Chapter 1, Section 1.1, Lemma 1]). We give its proof for the readers’ convenience.
Proof
By a density argument, it is sufficient to prove (B.3) for all \(\varphi \in C_c^\infty (U)\). We extend \(\varphi \) to \({\mathbb {R}}^2\) by setting it to be zero outside U. Then
for each \(s=(s_1,s_2)\in {\mathbb {R}}^2\). Thus Hölder’s inequality implies
Similarly, we obtain
From the above two inequalities we deduce that
We again use Hölder’s inequality to get
and a similar inequality for the last line of (B.4). By these inequalities and (B.4),
Since \(\varphi \in C_c^\infty (U)\) is compactly supported in U, this inequality implies (B.3). \(\square \)
Proof of Lemma 4.1
Since \(\Gamma \) is compact and without boundary, we can take a finite number of bounded open sets in \(\mathbb {R}^2\) and local parametrizations of \(\Gamma \)
such that \(\{\mu _k(U_k)\}_{k=1}^{k_0}\) is an open covering of \(\Gamma \). Let \(\{\xi _k\}_{k=1}^{k_0}\) be a partition of unity on \(\Gamma \) subordinate to \(\{\mu _k(U_k)\}_{k=1}^{k_0}\) consisting of \(C^1\) functions on \(\Gamma \). We may assume that \(\xi _k\) is supported in \(\mu _k({\mathcal {K}}_k)\) with some compact subset \({\mathcal {K}}_k\) of \(U_k\) for \(k = 1,\ldots , k_0\). For \(\eta \in H^1(\Gamma )\) and \(k = 1,\ldots , k_0\) let \((\xi _k\eta )^{\flat }:= (\xi _k\eta )\circ \mu _k\) on \(U_k\). Then \((\xi _k\eta )^{\flat }\in H_0^1(U_k)\) by Lemma (B.1), since \(\xi _k\eta \) belongs to \(H^1(\Gamma )\) and is supported in \(\mu _k(\mathcal {K}_k)\). Hence
by (B.3). We apply (B.1) and (B.2) to this inequality to get
Since this inequality holds for all \(k = 1,\ldots , k_0\), we obtain (4.1). \(\square \)
Next we present the proof of Lemma 4.3.
Proof of Lemma 4.3
To prove (4.6) we use the anisotropic Agmon inequality
for \(V=(0,1)^3\) and \(\Phi \in H^2(V)\) (see [54, Proposition 2.2]). For this purpose, we use a partition of unity on \(\Gamma \) to localize a function on \(\Omega _\varepsilon \).
Since \(\Gamma \) is compact and without boundary, we can take a finite number of bounded open sets in \({\mathbb {R}}^2\) and local parametrizations of \(\Gamma \)
such that \(\{\mu _k(U_k)\}_{k=1}^{k_0}\) is an open covering of \(\Gamma \). Let \(\{\eta _k\}_{k=1}^{k_0}\) be a partition of unity on \(\Gamma \) subordinate to \(\{\mu _k(U_k)\}_{k=1}^{k_0}\). We may assume that \(\eta _k\) is supported in \(\mu _k({\mathcal {K}}_k)\) with some compact subset \({\mathcal {K}}_k\) of \(U_k\) for each \(k=1,\dots ,k_0\). Let \({\bar{\eta }}_k:=\eta _k\circ \pi \) be the constant extension of \(\eta _k\) and
Then \(\{\zeta _k(V_k)\}_{k=1}^{k_0}\) is an open covering of \(\Omega _\varepsilon \) and \(\{{\bar{\eta }}_k\}_{k=1}^{k_0}\) is a partition of unity on \(\Omega _\varepsilon \) subordinate to \(\{\zeta _k(V_k)\}_{k=1}^{k_0}\). For \(\varphi \in H^2(\Omega _\varepsilon )\) we define
Then \(\varphi _k\) is supported in \(\zeta _k({\mathcal {K}}_k\times (0,1))\subset \zeta _k(V_k)\) and
by (4.3). Therefore, if we prove
for all \(k=1,\dots ,k_0\), then we obtain (4.6) for \(\varphi \).
Let us show (B.6). In what follows, we fix and suppress the index k. Hence we assume that \(\varphi \in H^2(\Omega _\varepsilon )\) is supported in \(\zeta ({\mathcal {K}}\times (0,1))\subset \zeta (V)\) with some compact subset \({\mathcal {K}}\) of U. Taking U small and scaling it, we may further assume that
The local parametrization \(\zeta \) of \(\Omega _\varepsilon \) is of the form
where \(\mu :U\rightarrow \Gamma \) is a \(C^5\) local parametrization of \(\Gamma \) and
Since \(g_0\), \(g_1\), and n are of class \(C^4\) on \(\Gamma \), \({\mathcal {K}}\) is compact in U, and \(h_\varepsilon \) is an affine function of \(s_3\), there exists a constant \(c>0\) independent of \(\varepsilon \) such that
Let \(\nabla _s\zeta \) be the gradient matrix of \(\zeta \) in \(s\in {\mathbb {R}}^3\), J the function given by (3.30), and \(\theta \) the Riemannian metric of \(\Gamma \) given by
Then
which we prove at the end of the proof. Moreover, since \(\det \theta \) is continuous and strictly positive on U and \({\mathcal {K}}\) is compact in U, we have
Applying this inequality, (2.1), and (3.31) to (B.11) we obtain
Now let \(\Phi :=\varphi \circ \zeta \) on V. Then
and \(\Phi \) is supported in \({\mathcal {K}}\times (0,1)\) since \(\varphi \) is supported in \(\zeta ({\mathcal {K}}\times (0,1))\). Also, since
we observe by (B.12) that
We differentiate \(\Phi (s)=\varphi (\zeta (s))\) in \(s\in V\). Then
for \(s\in V\) and \(i=1,2\), and
for \(s=(s',s_3)\in V\). Hence (B.9) and the boundedness of g on \(\Gamma \) imply that
for \(i,k=1,2\) and \(s\in {\mathcal {K}}\times (0,1)\). Since \(\Phi \) is supported in \({\mathcal {K}}\times (0,1)\), we deduce from these inequalities and (B.14) that
for \(i,k=1,2\). Similarly, for \(i,j=1,2,3\) with \(i\ne j\) we have
and thus \(\Phi \in H^2(V)\). Hence we can apply (B.5) to \(\Phi =\varphi \circ \zeta \) and use (B.13), (B.15), and (B.16) to obtain (B.6).
It remains to show the formula (B.11). In what follows, we use the notation
for a function \(\eta \) on \(\Gamma \). Note that, since \(\eta (\mu (s'))={\bar{\eta }}(\mu (s'))\) by \(\mu (s')\in \Gamma \),
for \(i=1,2\) by (3.5). By (B.7) and (B.8) we have
We differentiate \(\zeta (s)\) and apply (B.18) and \(-\nabla _\Gamma n=W=W^T\) on \(\Gamma \) to get
for \(s=(s',s_3)\in V\), where \(h_\varepsilon (s)\) is given by (B.8) and
From now on, we fix and suppress the arguments \(s'\) and s. By (B.19) we have
Here we consider \(n^\flat \in {\mathbb {R}}^3\) and \(\eta _\varepsilon :=(\eta _\varepsilon ^1,\eta _\varepsilon ^2)^T\in {\mathbb {R}}^2\) as column vectors. Since \(\partial _{s_1}\mu \) and \(\partial _{s_2}\mu \) are tangent to \(\Gamma \) at \(\mu (s')\) we have \((\nabla _{s'}\mu )n^\flat =0\). Moreover,
From these equalities and the symmetry of the matrix \(W^\flat \) it follows that
Hence by elementary row operations we have
Since \(\det [\nabla _s\zeta (\nabla _s\zeta )^T]=(\det \nabla _s\zeta )^2\), the above equality implies that
To compute the right-hand side we define \(3\times 3\) matrices
Then by \((\nabla _{s'}\mu )n^\flat =0\), \(W^\flat n^\flat =0\), the symmetry of \(W^\flat \), and (B.10) we have
Noting that A and \(I_3-h_\varepsilon W^\flat \) are \(3\times 3\) matrices, we use these equalities to get
where the last equality follows from \(\det (I_3-h_\varepsilon W^\flat )=J(\mu ,h_\varepsilon )\). From this equality and (B.20) we deduce that
This equality yields (B.11) since \(g^\flat \) and \(J(\mu ,h_\varepsilon )\) are positive by (2.1) and (3.31). \(\square \)
Finally, let us prove Lemma 7.4. To this end, we give an auxiliary result.
Lemma B.3
Let \(E_1\), \(E_2\), and \(E_3\) be vector fields on an open subset U of \({\mathbb {R}}^3\) such that \(\{E_1(x),E_2(x),E_3(x)\}\) is an orthonormal basis of \({\mathbb {R}}^3\) for each \(x\in U\) and
Then for \(u\in C^1(U)^3\) we have
Proof
By the assumption, \(\mathrm {curl}\,u=\sum _{i=1}^3(\mathrm {curl}\,u\cdot E_i)E_i\). Since \(E_1=E_2\times E_3\),
Calculating \(\mathrm {curl}\,u\cdot E_i\), \(i=2,3\) in the same way we obtain (B.21). \(\square \)
Proof of Lemma 7.4
Let \(u\in C^1(\Omega _\varepsilon )^3\) and \(u^a\) be given by (6.47). Since the surface \(\Gamma \) is compact and without boundary, we can take a finite number of relatively open subsets \(O_k\) of \(\Gamma \) and pairs of tangential vector fields \(\{\tau _1^k,\tau _2^k\}\) on \(O_k\), \(k=1,\dots ,k_0\) such that \(\Gamma =\bigcup _{k=1}^{k_0}O_k\), the triplet \(\{\tau _1^k,\tau _2^k,n\}\) forms an orthonormal basis of \({\mathbb {R}}^3\) on \(O_k\), and
for each \(k=1,\dots ,k_0\). Then since \(\Omega _\varepsilon =\bigcup _{k=1}^{k_0}U_k\) with
it is sufficient to show (7.5) in \(U_k\) for each \(k=1,\dots ,k_0\).
From now on, we fix and suppress the index k and carry out calculations in U unless otherwise stated. We apply (B.21) to \(u^a\) with
and use \(P\tau _i=\tau _i\) for \(i=1,2\) and \(Pn=0\) on O to get
By this equality, \(({\bar{n}}\cdot \nabla )u^a=\partial _nu^a\), and \(|{\bar{\tau }}_1|=|{\bar{\tau }}_2|=|{\bar{n}}|=1\) we get
Let us estimate each term on the right-hand side. By (4.3) and (6.47) we have
Hence it follows from (4.19) that
To estimate the other terms we set
so that \(u^a=u_\tau ^a+u_n^a\). Let \(i=1,2\). Since \(u_\tau ^a\cdot {\bar{n}}=0\) in U, we have
Hence by (3.17) and \(|{\bar{\tau }}_i|=1\) we get
Next we deal with \(({\bar{\tau }}_i\cdot \nabla )u_n^a\cdot {\bar{n}}\). Since \(\tau _i=P\tau _i\), \(P=P^T\), and \(|\tau _i|=1\) on O,
Moreover, by the definition (B.24) of \(u_n^a\) we have
and thus we deduce from (3.13), (3.17), (4.19), and (4.20) that
In the last inequality we also used \(M_\tau u=PMu\) on \(\Gamma \) and the \(C^4\)-regularity of P on \(\Gamma \). By (B.26) and (B.27) we observe that
Noting that \(u^a=u_\tau ^a+u_n^a\), we conclude by (B.22), (B.23), (B.25), and (B.28) that the inequality (7.5) holds in U. \(\square \)
Appendix C: Weaker Assumption on the Friction Coefficients
In this appendix we discuss the global existence of a strong solution to (1.2) under an assumption on \(\gamma _\varepsilon ^0\) and \(\gamma _\varepsilon ^1\) weaker than Assumption 2.1.
For a fixed \(\delta \in [0,1]\) we assume that there exists a constant \(c>0\) such that
for all \(\varepsilon \in (0,1]\) and that either of the condition (A2) of Assumption 2.2 or the following condition is satisfied:
-
(A4)
There exists a constant \(c>0\) such that
$$\begin{aligned} \gamma _\varepsilon ^0 \ge c\varepsilon ^\delta \quad \text {for all}\quad \varepsilon \in (0,1] \quad \text {or}\quad \gamma _\varepsilon ^1 \ge c\varepsilon ^\delta \quad \text {for all}\quad \varepsilon \in (0,1]. \end{aligned}$$
Then we can show the global existence of a strong solution to (1.2) under an additional condition on \(\delta \) as in the case of flat thin domains [12, 14]. To see this, let us explain which inequalities in the first part [34] and this paper are modified. In what follows, we omit the proofs of modified inequalities if they are obtained just by replacing the estimates for \(\gamma _\varepsilon ^0\) and \(\gamma _\varepsilon ^1\) in the proofs of the original ones.
As in Sect. 2, let \({\mathcal {H}}_\varepsilon :=L_\sigma ^2(\Omega _\varepsilon )\) and \({\mathcal {V}}_\varepsilon :={\mathcal {H}}_\varepsilon \cap H^1(\Omega _\varepsilon )^3\). First we note that we can take the same \(\varepsilon _0\in (0,1]\) as in [34, Theorem 2.4] to get the boundedness and coerciveness of the bilinear form \(a_\varepsilon \) on \({\mathcal {V}}_\varepsilon \). However, they are not uniform in \(\varepsilon \) and the basic inequalities for the Stokes operator \(A_\varepsilon \) given in [34, Lemma 2.5] change as follows.
Lemma C.1
For all \(\varepsilon \in (0,\varepsilon _0]\) and \(u\in {\mathcal {V}}_\varepsilon \) we have
and
Moreover, if \(u\in D(A_\varepsilon )\), then
For a vector field u on \(\Omega _\varepsilon \), the auxiliary vector field G(u) given by (7.2) satisfies
instead of the second inequality of (7.3) (see [34, Lemma 7.2]). Using this inequality we can show the following estimate as in the proof of [34, Theorem 2.6].
Lemma C.2
For all \(\varepsilon \in (0,\varepsilon _0]\) and \(u\in D(A_\varepsilon )\) we have
We also observe that the uniform a priori estimate for the vector Laplace operator shown in [34, Lemma 6.1] gets slightly worse.
Lemma C.3
For all \(\varepsilon \in (0,1]\) and \(u\in H^2(\Omega _\varepsilon )^3\) satisfying (4.8) we have
Using Lemmas C.1–C.3 and noting that \(A_\varepsilon =-\nu {\mathbb {P}}_\varepsilon \Delta \) and \({\mathbb {P}}_\varepsilon \) is the orthogonal projection from \(L^2(\Omega _\varepsilon )^3\) onto \({\mathcal {H}}_\varepsilon \), we obtain the following norm equivalence for \(A_\varepsilon \) instead of [34, Theorem 2.7] (also note that \(1\le \varepsilon ^{(\delta -1)/2}\le \varepsilon ^{\delta -1}\) by \(\delta -1\le 0\)).
Lemma C.4
For all \(\varepsilon \in (0,\varepsilon _0]\) and \(u\in D(A_\varepsilon )\) we have
Moreover, if the condition (A2) is imposed, then
and, if the condition (A4) is imposed,
In particular, under both of the conditions (A2) and (A4) we have
Now let us move to the results of this paper. When the inequalities (C.1) are valid, for \(p\in [1,\infty )\) and \(u\in W^{2,p}(\Omega _\varepsilon )^3\) satisfying (4.8) on \(\Gamma _\varepsilon ^0\) or on \(\Gamma _\varepsilon ^1\) we have
instead of (4.16). This modified inequality, however, does not affect the other parts of this paper since (4.16) was used only in the proof of (6.61) and we get the same result even if we replace (4.16) by the above inequality due to the fact that the Weingarten map W of \(\Gamma \) does not vanish in general. We also note that most of the estimates in [34] and this paper are worse than the corresponding estimates for flat thin domains given in [12, 14] because of the nonzero curvatures of the limit set \(\Gamma \).
Next we see that the estimate (7.1) for the trilinear term changes to the one in which an additional term appears due to the modified inequalities (C.4) and (C.5).
Lemma C.5
For any \(\alpha >0\) there exist constants \(c_\alpha ^1,c_\alpha ^2,c_\alpha ^3>0\) such that
for all \(\varepsilon \in (0,\varepsilon _0]\) and \(u\in D(A_\varepsilon )\). (In fact, \(c_\alpha ^1\) does not depend on \(\alpha \).)
Proof
The proof is the same as that of Lemma 7.1 except for the estimates of
with \(\omega =\mathrm {curl}\,u\) and \(\Phi =\omega \times u^a\). For \(J_2\) we apply (7.7) and (C.5) instead of (5.6). Then we get
where
We observe by (5.8) and Young’s inequality \(ab\le \alpha a^2+c_\alpha b^2\) that
Also, applying Young’s inequality \(ab\le \alpha a^4+c_\alpha b^{4/3}\) to \(K_2^\prime \) we have
Combining the above estimates we obtain (after replacing \(c\alpha \) by \(\alpha \))
For \(J_3^1\) we deduce from (7.7) and (C.4) instead of (7.3) that
with the same \(K_1^\prime \) and \(K_2^\prime \) as above. Thus (C.8) holds for \(J_3^1\) and (C.7) follows. \(\square \)
Now we make an additional assumption on \(\delta \), which is the same as those in [12, 14], to derive a good estimate for the trilinear term in terms of \(A_\varepsilon \).
Lemma C.6
Suppose that
Then there exist constants \(d_1,d_2,d_3>0\) such that
for all \(\varepsilon \in (0,\varepsilon _0]\) and \(u\in D(A_\varepsilon )\).
Proof
First we observe by (C.2), (C.3), and \(\varepsilon ^{(1-\delta )/2}\le 1\) that
We apply this inequality and (C.6) to (C.7) to get
with positive constants c, \(d_\alpha ^1\), \(d_\alpha ^2\), and \(d_\alpha ^3\) independent of \(\varepsilon \). Moreover, since
by (C.3), under the assumption (C.9) we have \(\sigma \ge 0\) and thus (note that \(\varepsilon \le 1\))
We also observe that \(\varepsilon ^{2\delta -2}=\varepsilon ^{-1+(2\delta -1)}\le \varepsilon ^{-1}\) since \(2\delta -1\ge 0\) by (C.9). Applying these inequalities to (C.11) and taking \(\alpha =1/4c\) we obtain (C.10). \(\square \)
Finally, we observe that the global existence of a strong solution to (1.2) can be established as in Theorem 2.6.
Theorem C.7
Suppose that the inequalities (C.1) are valid and that either of the conditions (A2) and (A4) is satisfied. Then under the condition (C.9) there exists a constant \(c_0\in (0,1]\) such that the following statement holds: for each \(\varepsilon \in (0,\varepsilon _0]\) suppose that the given data
satisfy
Then there exists a global-in-time strong solution
to the Navier–Stokes equations (1.2).
Note that (C.12) implies \(\Vert u_0^\varepsilon \Vert _{L^2(\Omega _\varepsilon )}^2\le c_0\varepsilon ^{-\delta }\) with \(\delta \le 1\), which is slightly worse than the condition on the \(L^2(\Omega _\varepsilon )\)-norm of \(u_0^\varepsilon \) in (2.11).
Proof
We only show the points modified from the proof of Theorem 2.6. First we deduce from (C.2) and (C.12) that
as in (8.6). Also, (8.8) holds by (8.7), (C.12), and \(\delta \le 1\). Thus we can show (8.15) for the strong solution \(u^\varepsilon \) on \([0,T_{\max })\) as in the proof of Theorem 2.6.
Next we prove (8.16) by contradiction. Assume to the contrary that there exists \(T\in (0,T_{\max })\) such that (8.17) and (8.18) hold. Then as in (8.20) we have
by (C.10) and Young’s inequality, where \(\xi \) and \(\zeta \) are given by (8.21) and
For this additional function \(\psi \), we observe by (8.15) and \(c_0\le 1\) that
for all \(t\in (0,T]\) and, if \(T\ge 1\) and \(t\in [1,T]\),
Therefore, proceeding as in the proof of Theorem 2.6 with \(\zeta \) replaced by \(\zeta +\psi \), we can get a contradiction and conclude that (8.16) is valid, which yields \(T_{\max }=\infty \). \(\square \)
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Miura, TH. Navier–Stokes Equations in a Curved Thin Domain, Part II: Global Existence of a Strong Solution. J. Math. Fluid Mech. 23, 7 (2021). https://doi.org/10.1007/s00021-020-00534-2
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DOI: https://doi.org/10.1007/s00021-020-00534-2