In this section, we consider the following problem:
$$\begin{aligned} \textstyle\begin{cases} -\nu \Delta {\mathbf {v}}+\nabla p=\mathbf{f}, \qquad \nabla \cdot { \mathbf {v}}=g \quad \text{in } \Omega , \\ 2\nu \Pi {\mathbf{D}}({\mathbf {v}})\mathbf{n}+\gamma \nabla _{\Gamma }\rho _{1}={ \mathbf{a}}, \qquad {\mathbf {v}}\cdot {\mathbf{n}}=a_{3}, \\ -\kappa _{1}\nabla _{\Gamma }^{2}\rho _{1}+\chi _{1}\rho _{1}+\nabla _{\Gamma }\cdot {\mathbf {v}}=h_{1}\quad \text{on } \Gamma , \\ 2\nu \Pi {\mathbf{D}}({\mathbf {v}})\mathbf{n}+\beta \Pi {\mathbf {v}}+ \frac{\gamma }{2}\nabla _{\Sigma }\rho _{2}=\mathbf{b}, \qquad {\mathbf {v}} \cdot {\mathbf{n}}=b_{3}, \\ -\kappa _{2}\nabla _{\Sigma }^{2}\rho _{2}+\chi _{2}\rho _{2}+ \frac{1}{2}\nabla _{\Sigma }\cdot {\mathbf {v}}=h_{2} \quad \text{on } \Sigma , \\ ({\mathbf {v}}-\kappa _{1}\nabla _{\Gamma }\rho _{1})\cdot {\mathbf{e}}_{1}=m_{1}, \\ (\frac{1}{2}{\mathbf {v}}-\kappa _{2}\nabla _{\Sigma }\rho _{2})\cdot { \mathbf{e}}_{2}=m_{2} \quad \text{on } M, \\ {\mathbf {v}}\longrightarrow {\mathbf {0}}, \qquad\rho _{2} \longrightarrow 0 \quad\text{as } x_{3}\longrightarrow -\infty , \end{cases}\displaystyle \end{aligned}$$
(32)
where ν, γ, \(\kappa _{1}\), \(\kappa _{2}\), β, \(\chi _{1}\), and \(\chi _{2}\) are positive constants, and Ω is defined for a given function \(h \in C^{3+\alpha }_{s+1}(B)\) in the same manner as \(\Omega _{h}\) was defined in Sect. 2. Similarly, Γ, Σ, M, n, \(\mathbf{e}_{1}\), \(\mathbf{e}_{2}\), \(\nabla _{\Gamma }\), and \(\nabla _{\Sigma }\) are defined in the same way as in Sect. 2. We assume that the domain Ω and the data f, g, a, \(a_{3}\), b, \(b_{3}\), \(h_{1}\), \(h_{2}\), \(m_{1}\), and \(m_{2}\) do not depend on the rotation angle ϕ around the \(x_{3}\)-axis, and the ϕ-components of f, a, and b vanish in the cylindrical coordinates \((r,\phi ,z)\) defined by \((x_{1},x_{2},x_{3})=(r\cos \phi ,r\sin \phi ,z)\). We denote the contact angle between Γ and Σ as θ and assume that \(\theta \neq0, \pi \).
The following is the main result in this section.
Theorem 2
Let \(0<\alpha <1\) and s be a constant satisfying the conditions \(s\leq 2+\alpha \), \(s\notin {\mathbb{Z}}\), and \(0< s<\min (\lambda _{0},\pi /\theta )\), where \(\lambda _{0}\) is the constant given in Lemma 1. Assume that \(\mathbf{f}\in C^{\alpha ,\mu }_{s-2}(\Omega ,M)\), \({g}\in C^{1+\alpha ,\mu }_{s-1}(\Omega ,M)\), \(\mathbf{a}\in C^{1+\alpha }_{s-1}(\Gamma ,M)\), \(a_{3}\in C^{2+\alpha }_{s}(\Gamma ,M)\), \(\mathbf{b}\in C^{1+\alpha ,\mu }_{s-1}(\Sigma ,M)\), \(b_{3}\in C^{2+\alpha ,\mu }_{s}(\Sigma ,M)\), \(h_{1}\in C^{1+\alpha }_{s-1}(\Gamma ,M)\), \(h_{2}\in C^{1+\alpha ,\mu }_{s-1}(\Sigma ,M)\), and \(m_{1}, m_{2}\in C^{s}(M)\). In addition, the following condition holds:
$$\begin{aligned} \int _{\Omega }g \,dx= \int _{\Gamma }a_{3} \,dS+ \int _{\Sigma }b_{3} \,dS. \end{aligned}$$
(33)
For the case \(\theta <\frac{\pi }{2}\), we also assume the compatibility condition at the contact line M:
$$\begin{aligned} 0={}&\mathbf{b}\cdot {\mathbf{e}}_{2}-\beta {\mathbf {v}}|_{M}\cdot {\mathbf{e}}_{2}+ \frac{\gamma }{2\kappa _{2}} \biggl(m_{2}-\frac{1}{2}{\mathbf {v}}|_{M}\cdot { \mathbf{e}}_{2}\biggr)+\mathbf{a}\cdot {\mathbf{e}}_{1} \\ &{}+\frac{\gamma }{\kappa _{1}}(m_{1}-{\mathbf {v}}|_{M}\cdot { \mathbf{e}}_{1})+2 \nu \partial _{z}b_{3}+2\nu \biggl(\frac{\partial a_{3}}{\partial {\tau }}-{ \mathbf {v}}|_{M}\cdot \frac{\partial {\mathbf{n}}}{\partial {\tau }} \biggr) \end{aligned}$$
(34)
with
$$\begin{aligned} {\mathbf {v}}|_{M}=(v_{r},v_{\phi },v_{z})|_{M}= \bigl(b_{3},v_{\phi },h'(1)b_{3}+ \sqrt{1+h'(1)^{2}}a_{3}\bigr), \end{aligned}$$
where τ is given by \((\frac{1}{\sqrt{1+h'(r)^{2}}},0,\frac{h'(r)}{\sqrt{1+h'(r)^{2}}})\) in the cylindrical coordinates introduced above. Then there exists a constant \(\mu >0\) such that problem (32) has a unique axisymmetric solution with the following properties: the ϕ-component of v vanishes, \({\mathbf {v}}\in C^{2+\alpha ,\mu }_{s}(\Omega ,M)\), \(\nabla p\in C^{\alpha ,\mu }_{s-2}(\Omega ,M)\), \(\rho _{1}\in C^{3+\alpha }_{s+1}(\Gamma ,M)\), and \(\rho _{2}\in C^{3+\alpha ,\mu }_{s+1}(\Sigma ,M)\), and these functions satisfy the estimate
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha ,\mu )}_{s,\Omega ,M}+ \vert \nabla p \vert ^{(\alpha , \mu )}_{s-2,\Omega ,M}+ \vert \rho _{1} \vert ^{(3+\alpha )}_{s+1,\Gamma ,M}+ \vert \rho _{2} \vert ^{(3+\alpha ,\mu )}_{s+1,\Sigma ,M} \\ &\quad\leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha ,\mu )}_{s-2,\Omega ,M}+ \vert g \vert ^{(1+ \alpha ,\mu )}_{s-1,\Omega ,M}+ \vert {\mathbf{a}} \vert ^{(1+\alpha )}_{s-1,\Gamma ,M}+ \vert a_{3} \vert ^{(2+ \alpha )}_{s,\Gamma ,M}+ \vert {\mathbf{b}} \vert ^{(1+\alpha ,\mu )}_{s-1,\Sigma ,M} \\ &\qquad{} + \vert b_{3} \vert ^{(2+\alpha ,\mu )}_{s,\Sigma ,M}+ \vert h_{1} \vert ^{(1+ \alpha )}_{s-1,\Gamma ,M}+ \vert h_{2} \vert ^{(1+\alpha ,\mu )}_{s-1,\Sigma ,M}+ \vert m_{1} \vert ^{(s)}_{M}+ \vert m_{2} \vert ^{(s)}_{M} \bigr) \end{aligned}$$
(35)
for some constant \(C>0\) that is independent of the data.
We begin with the following lemma.
Lemma 4
Under the assumption stated in Theorem 2for g, \(a_{3}\), and \(b_{3}\), there exists a vector field \({\mathbf {w}}\in C^{2+\alpha ,\mu }_{s}(\Omega ,M)\) satisfying
$$\begin{aligned} \textstyle\begin{cases} \nabla \cdot {\mathbf {w}}=g \quad\textit{in } \Omega , \\ {\mathbf {w}}\cdot {\mathbf{n}}=a_{3} \quad\textit{on } \Gamma , \qquad { \mathbf {w}}\cdot {\mathbf{n}}=b_{3} \quad\textit{on } \Sigma \end{cases}\displaystyle \end{aligned}$$
(36)
and the estimate
$$\begin{aligned} & \vert {\mathbf {w}} \vert ^{(2+\alpha ,\mu )}_{s, \Omega ,M} \leq C \bigl( \vert g \vert ^{(1+ \alpha ,\mu )}_{s-1, \Omega ,M}+ \vert a_{3} \vert ^{(2+\alpha )}_{s, \Gamma ,M}+ \vert b_{3} \vert ^{(2+ \alpha ,\mu )}_{s, \Sigma ,M} \bigr) \end{aligned}$$
(37)
for a constant \(C>0\) that is independent of the data.
Proof
We construct the function in the form \({\mathbf {w}}=\nabla \Phi \), where Φ is a solution of
$$\begin{aligned} \textstyle\begin{cases} \Delta \Phi = g \quad\text{in } \Omega , \\ {\frac{\partial \Phi }{\partial n}|_{\Gamma }=a_{3},\qquad \frac{\partial \Phi }{\partial n}|_{\Sigma }=b_{3}}. \end{cases}\displaystyle \end{aligned}$$
(38)
We first construct a function \(\Phi '\) satisfying conditions (38)2,3 and the estimate
$$\begin{aligned} \bigl\vert \Phi ' \bigr\vert ^{(3+\alpha ,\mu )}_{s+1,\Omega ,M} \leq C\bigl( \vert a_{3} \vert ^{(2+\alpha )}_{s, \Gamma ,M}+ \vert b_{3} \vert ^{(2+\alpha ,\mu )}_{s,\Sigma ,M}\bigr). \end{aligned}$$
Then we introduce the new unknown \(\Phi -\Phi '\) to reduce problem (38) to one in which \(a_{3}=b_{3}=0\). In the following argument, we again denote \(\Phi -\Phi '\) and \(g-\Delta \Phi '\) by Φ and g, respectively.
Now, let us set
$$\begin{aligned} D^{1}_{2}(\Omega )=\bigl\{ u\in L_{2,\mathrm{loc}}(\Omega ) | \partial _{x} u \in L_{2}( \Omega )\bigr\} . \end{aligned}$$
For \(u\in D^{1}_{2}(\Omega )\), we set
$$\begin{aligned}{} [u]= \bigl\{ w\in D^{1}_{2}(\Omega ) | w=u+c \text{ for a constant } c \bigr\} , \end{aligned}$$
and let \(\dot{D}^{1}_{2}(\Omega )\) be the space of all equivalence classes \([u]\). As shown in [6], the space \(\dot{D}^{1}_{2}(\Omega )\) is a Hilbert space with the scalar product
$$\begin{aligned} (u,v)= \int _{\Omega }\nabla u\cdot \nabla v \,dx. \end{aligned}$$
We prove the weak solvability of problem (38) with \(a_{3}=b_{3}=0\) in class \(\dot{D}^{1}_{2}(\Omega )\) for arbitrary \(g \in L_{2}(\Omega )\). Let \(\zeta _{k}\) (\(k\in \mathbb{N}\)) be the cut-off function such that
$$\begin{aligned} \zeta _{k}(x)=\textstyle\begin{cases} 1& (x_{3}\geq -k), \\ 0& (x_{3}\leq -(k+1)). \end{cases}\displaystyle \end{aligned}$$
Set \(g_{k}=\zeta _{k} g\) and let \(\Phi _{k}\) be the corresponding solution of the problem with \(g=g_{k}\). We multiply the equation \(\Delta \Phi _{k} =g_{k}\) by \(\phi \in \dot{D}^{1}_{2}(\Omega )\) and integrate both sides to obtain
$$\begin{aligned} - \int _{\Omega }\nabla \Phi _{k}\cdot \nabla \phi \,dx = \int _{\Omega }g_{k} \phi \,dx. \end{aligned}$$
(39)
The Riesz representation theorem then implies that there exists a unique solution \(\Phi _{k} \in \dot{D}^{1}_{2}(\Omega )\) of (39), and we obtain the estimate
$$\begin{aligned} \Vert \nabla \Phi _{k} \Vert _{2,\Omega }\leq C \Vert g_{k} \Vert _{2,\Omega }. \end{aligned}$$
Thus, by taking the limit \(k\to \infty \) in (39), we have a solution \(\Phi \in \dot{D}^{1}_{2}(\Omega )\) satisfying the estimate
$$\begin{aligned} \Vert \nabla \Phi \Vert _{2,\Omega }\leq C \Vert g \Vert _{2,\Omega }. \end{aligned}$$
For problem (38) with \(a_{3}=b_{3}=0\), we can obtain a decay estimate that is similar to (56). Based on this estimate, we can obtain estimate (37) using a method of localization similar to that used to obtain estimates (71) and (73). Because of the axial symmetry of the data and the domain, the solution constructed above is axially symmetric. Therefore, we can perform the above localization procedure for a two-dimensional problem using estimate (41) given below. In Lemma 5, \(d_{\theta }\), \(\gamma _{\theta }\), \(\gamma _{0}\), and M are defined in the manner specified in Sect. 4. As the result is well known, the proof is omitted.
Lemma 5
Assume that \(0<\alpha <1\), \(s\notin {\mathbb{Z}}\), \(0< s<\pi /\theta \), and \(s\leq 2+\alpha \). Further, assume that \(g\in C^{1+\alpha }_{s-1}(d_{\theta },M)\), \(a_{3}\in C^{2+\alpha }_{s}(\gamma _{\theta },M)\), and \(b_{3}\in C^{2+\alpha }_{s}(\gamma _{0},M)\), and that their supports are compact. Additionally, assume that the condition
$$\begin{aligned} \int _{d_{\theta }} g \,dx= \int _{\gamma _{\theta }} a_{3} \,dS+ \int _{\gamma _{0}} b_{3} \,dS \end{aligned}$$
(40)
is satisfied. Then problem (38) with \(\Omega =d_{\theta }\), \(\Gamma =\gamma _{\theta }\), and \(\Sigma =\gamma _{0}\) has a unique solution \(\nabla \Phi \in C^{2+\alpha }_{s}(d_{\theta },M)\) that satisfies the estimate
$$\begin{aligned} \vert \nabla \Phi \vert ^{(2+\alpha )}_{s,d_{\theta },M}\leq C \bigl( \vert g \vert ^{(1+\alpha )}_{s-1, \gamma _{\theta },M}+ \vert a_{3} \vert ^{(2+\alpha )}_{s,\gamma _{\theta },M}+ \vert b_{3} \vert ^{(2+ \alpha )}_{s,\gamma _{0},M}\bigr) \end{aligned}$$
(41)
for a constant \(C>0\) that is independent of the data.
Thus, we have proved Lemma 4. □
Lemma 6
Under the assumptions stated in Theorem 2for \(h_{1}\), \(h_{2}\), \(m_{1}\), and \(m_{2}\), there exists a unique solution \((s_{1},s_{2})\) to the problem
$$\begin{aligned} \textstyle\begin{cases} -\kappa _{1}\nabla _{\Gamma }^{2}s_{1}+\chi _{1}s_{1}=h_{1} & \textit{on } \Gamma , \\ -\kappa _{2}\nabla _{\Sigma }^{2}s_{2}+\chi _{2}s_{2}=h_{2} & \textit{on } \Sigma , \\ -\kappa _{1}\nabla _{\Gamma }s_{1}\cdot {\mathbf{e}}_{1}=m_{1},\qquad{-}\kappa _{2}\nabla _{\Sigma }s_{2}\cdot {\mathbf{e}}_{2}=m_{2} & \textit{on } M \end{cases}\displaystyle \end{aligned}$$
(42)
satisfying the estimate
$$\begin{aligned} & \vert s_{1} \vert ^{(3+\alpha )}_{s+1,\Gamma ,M}+ \vert s_{2} \vert ^{(3+\alpha ,\mu )}_{s+1, \Sigma ,M} \\ &\quad\leq C \bigl( \vert h_{1} \vert ^{(1+\alpha )}_{s-1,\Gamma ,M}+ \vert h_{2} \vert ^{(1+ \alpha ,\mu )}_{s-1,\Sigma ,M}+ \vert m_{1} \vert ^{(s)}_{M}+ \vert m_{2} \vert ^{(s)}_{M} \bigr) \end{aligned}$$
(43)
for a constant \(C>0\) that is independent of the data.
Proof
The proof is similar to that of Lemma 4. The existence of a weak solution of problem (42) is easily shown. In addition, for the problem
$$\begin{aligned} \textstyle\begin{cases} -\Delta \rho +\chi \rho = h \quad \text{in } \mathbb{R}^{2}_{+} \equiv \{(z_{1},z_{2})\in \mathbb{R}^{2} | z_{2}>0\}, \\ \partial _{z_{2}} \rho |_{z_{2}=0} = H,\quad \rho \to 0\ (z_{2}\to \infty ), \end{cases}\displaystyle \end{aligned}$$
where \(\chi \geq 0\) is a constant, we can obtain the following estimate in the same manner as in the proof of Theorem 7.1 in [30] under the assumption that the supports of the data are compact:
$$\begin{aligned} \vert \rho \vert ^{(3+\alpha )}_{s+1,\mathbb{R}^{2}_{+},\mathbb{R}}\leq C \bigl( \vert h \vert ^{(1+ \alpha )}_{s-1,\mathbb{R}^{2}_{+},\mathbb{R}}+ \vert H \vert ^{(s)}_{\mathbb{R}} \bigr). \end{aligned}$$
(44)
Using this estimate, in addition to a decay estimate for \(s_{1}, s_{2}\) similar to that in (56), we achieve the desired result in a similar manner to that used to obtain estimates (71) and (73). □
With the aid of Lemmas 4 and 6, problem (32) reduces to the problem with \(g=a_{3}=b_{3}=h_{1}=h_{2}=m_{1}=m_{2}=0\). Therefore, we hereafter consider (32) under this assumption.
Let us prove the weak solvability of the problem. After multiplying (32)1 by \({\mathbf {v}}'\), (32)5 by \(\rho _{1}'\), and (32)8 by \(\rho _{2}'\), we use integration by parts to obtain
$$\begin{aligned} &2\nu \int _{\Omega }{\mathbf{D}}({\mathbf {v}}): \mathbf{D}\bigl({ \mathbf {v}}'\bigr)\,dx + \beta \int _{\Sigma }{\mathbf {v}}\cdot {\mathbf {v}}' \,dS + \kappa _{1} \gamma \int _{\Gamma }\nabla _{\Gamma }\rho _{1}\cdot \nabla _{\Gamma }\rho _{1}' \,dS \\ &\qquad{}+\chi _{1}\gamma \int _{\Gamma }\rho _{1} \rho _{1}' \,dS +\kappa _{2} \gamma \int _{\Sigma }\nabla _{\Sigma }\rho _{2}\cdot \nabla _{\Sigma }\rho _{2}' \,dS+\chi _{2}\gamma \int _{\Sigma }\rho _{2} \rho _{2}' \,dS \\ &\qquad{} +\gamma \int _{\Gamma }\bigl(\nabla _{\Gamma }\rho _{1} \cdot {\mathbf {v}}'- \nabla _{\Gamma }\rho _{1}' \cdot {\mathbf {v}}\bigr) \,dS +\frac{\gamma }{2} \int _{\Sigma }\bigl(\nabla _{\Sigma }\rho _{2} \cdot {\mathbf {v}}'-\nabla _{\Sigma }\rho _{2}' \cdot {\mathbf {v}}\bigr) \,dS \\ &\quad= \int _{\Omega }{\mathbf{f}}\cdot {\mathbf {v}}' \,dx+ \int _{\Gamma }{\mathbf{a}} \cdot {\mathbf {v}}'\,dS+ \int _{\Sigma }{\mathbf{b}}\cdot {\mathbf {v}}'\,dS. \end{aligned}$$
(45)
Let \(J(\Omega )\) be the function space defined by
$$\begin{aligned} J(\Omega )\equiv \bigl\{ \mathbf{f}\in W^{1}_{2}(\Omega ) | \nabla \cdot {\mathbf{f}}=0, \mathbf{f}\cdot {\mathbf{n}}|_{\Gamma \cup \Sigma }=0 \bigr\} . \end{aligned}$$
We now prove the following inequality. The main difficulty is in the estimation of \(\|{\mathbf {v}}\|_{2,\Omega }\). A similar inequality in an infinite strip domain is derived in [18]. Because the domain is three-dimensional in the present case, more complicated arguments are necessary.
Lemma 7
For arbitrary \({\mathbf {v}}\in J(\Omega )\), the following inequality holds:
$$\begin{aligned} \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2,\Omega } \bigr)^{2}\leq C \int _{\Omega }{\mathbf{D}}({ \mathbf {v}}):\mathbf{D}({\mathbf {v}})\,dx. \end{aligned}$$
(46)
Proof
For any \({\mathbf {v}}\in J(\Omega )\), with the aid of Korn’s inequality, we have
$$\begin{aligned} \Vert \partial _{x}{\mathbf {v}} \Vert _{2,\Omega }^{2} \leq C \biggl( \int _{\Omega }{ \mathbf{D}}({\mathbf {v}}):\mathbf{D}({\mathbf {v}})\,dx+ \Vert {\mathbf {v}} \Vert ^{2}_{2, \Gamma } \biggr). \end{aligned}$$
As Γ is compact, using a similar argument as in the proof of Lemma 4 in [33] implies that there exists a constant \(C(\epsilon )\) satisfying the inequality
$$\begin{aligned} \Vert {\mathbf {v}} \Vert ^{2}_{2,\Gamma }\leq \epsilon \Vert \partial _{x} { \mathbf {v}} \Vert _{2,\Omega }^{2}+C( \epsilon ) \int _{\Omega }{\mathbf{D}}({ \mathbf {v}}):\mathbf{D}({\mathbf {v}})\,dx \end{aligned}$$
for arbitrary \(\epsilon >0\). Combining these inequalities, we have
$$\begin{aligned} \Vert \partial _{x}{\mathbf {v}} \Vert ^{2}_{2,\Omega }\leq C \int _{\Omega }{\mathbf{D}}({ \mathbf {v}}):\mathbf{D}({\mathbf {v}})\,dx. \end{aligned}$$
(47)
Now, from the conditions \(\nabla \cdot {\mathbf {v}}=0\) and \({\mathbf {v}}\cdot {\mathbf{n}}|_{\Sigma }=0\), \(\int _{B} v_{3}(x',x_{3})\,dx'=0\) holds for arbitrary \(x_{3}<0\). From this, we have
$$\begin{aligned} \bigl\Vert v_{3}(\cdot ,x_{3}) \bigr\Vert _{2,B}\leq C \bigl\Vert \nabla _{x'}v_{3}(\cdot ,x_{3}) \bigr\Vert _{2,B}. \end{aligned}$$
(48)
By integrating the identity with \(n=2\),
$$\begin{aligned} \sum_{i,j=1}^{n} \bigl\{ \partial _{x_{i}}(v_{i}x_{j}v_{j})-( \partial _{x_{i}}v_{i})x_{j}v_{j}-v_{i}x_{j}( \partial _{x_{i}}v_{j}) \bigr\} - \vert {\mathbf {v}} \vert ^{2}=0 \end{aligned}$$
(49)
on B under the condition that \({\mathbf {v}}\cdot {\mathbf{n}}|_{\partial B}=0\), we obtain
$$\begin{aligned} \bigl\Vert {\mathbf {v}}'(\cdot ,x_{3}) \bigr\Vert _{2,B}\leq C \bigl\Vert \partial _{x'}{\mathbf {v}}'( \cdot ,x_{3}) \bigr\Vert _{2,B}, \end{aligned}$$
(50)
where \({\mathbf {v}}'=(v_{1},v_{2})\). We integrate (48) and (50) with respect to \(x_{3}\) to obtain
$$\begin{aligned} \Vert {\mathbf {v}} \Vert _{2,\Omega _{1}}\leq C \Vert \partial _{x}{\mathbf {v}} \Vert _{2, \Omega _{1}}. \end{aligned}$$
(51)
Next, we integrate (49) with \(n=3\) in the domain \(\Omega _{0,1}\) and find that
$$\begin{aligned} \Vert {\mathbf {v}} \Vert _{2,\Omega _{0,1}}\leq C \bigl( \Vert \partial _{x}{ \mathbf {v}} \Vert _{2,\Omega _{0,1}}+ \bigl\Vert {\mathbf {v}}(\cdot ,-1) \bigr\Vert _{2,B} \bigr), \end{aligned}$$
(52)
and we note the estimate
$$\begin{aligned} \bigl\Vert {\mathbf {v}}(\cdot ,-1) \bigr\Vert _{2,B}\leq C \Vert {\mathbf {v}} \Vert ^{(1)}_{2, \Omega _{1}} \end{aligned}$$
from (51) and (52) to obtain
$$\begin{aligned} \Vert {\mathbf {v}} \Vert _{2,\Omega }\leq C \Vert \partial _{x}{\mathbf {v}} \Vert _{2, \Omega }. \end{aligned}$$
(53)
Estimate (46) is obtained by combining inequalities (47) and (53). □
We now introduce the Lax–Milgram theorem.
Theorem 3
(Lax–Milgram [14])
Let H be a real Hilbert space with the norm \(\|\cdot \|\) and \(B: H\times H\to \mathbb{R}\) be a bilinear mapping. If there exist constants \(\alpha , \beta >0\) such that
$$\begin{aligned} &\bigl|B[u,v]\bigr|\leq \alpha \|u\|\|v\| \quad (u,v\in H), \\ &\beta \|u\|^{2}\leq B[u,u] \quad (u\in H). \end{aligned}$$
Then, for every linear functional \(f: H\to \mathbb{R}\), there exists a unique element \(u\in H\) such that \(B[u,v]=f(v)\) for all \(v\in H\).
Lemma 7 indicates that the bilinear form defined by the left-hand side of (45), which we denote as \(B[({\mathbf {v}},\rho _{1},\rho _{2}),({\mathbf {v}}',\rho _{1}',\rho _{2}')] \), satisfies the condition
$$\begin{aligned} B\bigl[({\mathbf {v}},\rho _{1},\rho _{2}),({ \mathbf {v}},\rho _{1},\rho _{2})\bigr] \geq C \bigl\{ \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2,\Omega }\bigr)^{2}+ \bigl( \Vert \rho _{1} \Vert ^{(1)}_{2, \Gamma } \bigr)^{2}+\bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,\Sigma }\bigr)^{2} \bigr\} . \end{aligned}$$
(54)
Thus, from Theorem 3, we have the following.
Theorem 4
For arbitrary \((\mathbf{f}, \mathbf{a}, \mathbf{b}) \in L_{2}(\Omega )\times L_{2}(\Gamma ) \times L_{2}(\Sigma )\), there exists a unique solution \(({\mathbf {v}},\rho _{1},\rho _{2})\in J(\Omega )\times W^{1}_{2}( \Gamma )\times W^{1}_{2}(\Sigma )\) satisfying (45) for all \(({\mathbf {v}}',\rho '_{1},\rho '_{2})\in J(\Omega )\times W^{1}_{2}( \Gamma )\times W^{1}_{2}(\Sigma )\) and the estimate
$$\begin{aligned} & \Vert {\mathbf {v}} \Vert ^{(1)}_{2,\Omega }+ \Vert \rho _{1} \Vert ^{(1)}_{2,\Gamma }+ \Vert \rho _{2} \Vert ^{(1)}_{2,\Sigma }\leq C \bigl( \Vert { \mathbf{f}} \Vert _{2,\Omega }+ \Vert { \mathbf{a}} \Vert _{2,\Gamma }+ \Vert {\mathbf{b}} \Vert _{2,\Sigma } \bigr) \end{aligned}$$
(55)
for a constant \(C>0\) that is independent of the data.
For the solution \(({\mathbf {v}},\rho _{1},\rho _{2})\) obtained above, the pressure p is determined from the following equation defined for arbitrary \({\mathbf {\eta }} \in H(\Omega )\equiv \{ \mathbf{f}\in W^{1}_{2}( \Omega ) | \mathbf{f}\cdot {\mathbf{n}}|_{\Gamma \cup \Sigma }=0 \} \) with a compact support:
$$\begin{aligned} &2\nu \int _{\Omega }{\mathbf{D}}({\mathbf {v}}): \mathbf{D}({\mathbf { \eta }})\,dx + \beta \int _{\Sigma }{\mathbf {v}}\cdot {\mathbf {\eta }} \,dS- \int _{\Omega }p \nabla \cdot {\mathbf {\eta }}\,dx \\ &\quad= \int _{\Omega }{\mathbf{f}}\cdot {\mathbf {\eta }}\,dx+ \int _{\Gamma }(\mathbf{a}- \gamma \nabla _{\Gamma }\rho _{1})\cdot {\mathbf {\eta }}\,dS+ \int _{\Sigma }\biggl(\mathbf{b}-\frac{\gamma }{2}\nabla _{\Sigma }\rho _{2}\biggr)\cdot { \mathbf {\eta }}\,dS. \end{aligned}$$
If we take the vector η such that
$$\begin{aligned} \nabla \cdot {\mathbf {\eta }} = p \quad\text{in } \Omega ',\qquad { \mathbf { \eta }}={\mathbf {0}} \quad\text{in } \Omega \backslash \Omega ',\qquad \Vert \partial _{x}{\mathbf {\eta }} \Vert _{2,\Omega }\leq C \Vert p \Vert _{2, \Omega '}, \end{aligned}$$
where \(\Omega '\subset \Omega \) is a bounded domain (for the construction of η, see, e.g., [13]), we have the estimate
$$\begin{aligned} \Vert p \Vert _{2,\Omega '}\leq C \bigl( \Vert {\mathbf{f}} \Vert _{2,\Omega }+ \Vert {\mathbf{a}} \Vert _{2, \Gamma }+ \Vert { \mathbf{b}} \Vert _{2,\Sigma } \bigr). \end{aligned}$$
Here, p is normalized by the condition \(\int _{\Omega '}p\,dx=0\).
The axial symmetry of the solution constructed above readily follows from the symmetry of the data and the domain. In addition, when \(f_{\phi }=a_{\phi }=b_{\phi }=0\), if we take \(v_{\phi }{\mathbf{e}}_{\phi }\) (\({ \mathbf{e}}_{\phi }=(-x_{2}/|x'|,x_{1}/|x'|,0)\)) as \({\mathbf {v}}'\) in (45) and take 0 as \(\rho _{1}'\) and \(\rho _{2}'\), we have the equality
$$\begin{aligned} \frac{\nu }{2} \int _{\Omega } \biggl(\frac{\partial v_{\phi }}{\partial z} \biggr)^{2}+ \biggl(\frac{\partial v_{\phi }}{\partial r}- \frac{v_{\phi }}{r} \biggr)^{2}r\,dr\,d\phi \,dz+ \beta \int _{\Sigma }v_{\phi }^{2} \,d\phi \,dz=0, \end{aligned}$$
which indicates that \(v_{\phi }=0\).
The following decay estimate is obtained in a similar manner to the proofs of Theorem 5.3 in [27] and Lemma 1 in [19].
Lemma 8
Let \(\Omega (\lambda ,t)=\{x\in \Omega | \lambda -t < x_{3}<\lambda +t \}\) and \(\Sigma (\lambda ,t)=\{x\in \Sigma | \lambda -t< x_{3}<\lambda +t \}\ (\lambda <-4)\). Assume that f, a, and b possess the same regularity assumed in Theorem 2. Then the solutions v and \(\rho _{2}\) of (45) satisfy the inequality
$$\begin{aligned} & \Vert {\mathbf {v}} \Vert _{2,\Omega (\lambda ,1)}^{(1)}+ \Vert \rho _{2} \Vert ^{(1)}_{2, \Sigma (\lambda ,1)} \leq Ce^{\mu \lambda } \bigl( \vert {\mathbf{f}} \vert _{s-2, \Omega ,M}^{(\alpha ,\mu )}+ \vert {\mathbf{a}} \vert ^{(1+\alpha )}_{s-1,\Gamma ,M}+ \vert { \mathbf{b}} \vert ^{(1+\alpha ,\mu )}_{s-1,\Sigma ,M} \bigr) \end{aligned}$$
(56)
for constants \(\mu >0\) and \(C>0\) that are independent of the data.
Proof
We use integration by parts in the domain \(\Omega (\lambda ,t)\) in a manner similar to that for (45) to obtain
$$\begin{aligned} &2\nu \int _{\Omega (\lambda ,t)} {\mathbf{D}}({\mathbf {v}}): \mathbf{D}({ \mathbf {v}})\,dx + \beta \int _{\Sigma (\lambda ,t)} \vert {\mathbf {v}} \vert ^{2} \,dS \\ &\qquad{}+\gamma \kappa _{2} \int _{\Sigma (\lambda ,t)} \vert \nabla _{\Sigma }\rho _{2} \vert ^{2} \,dS+\gamma \chi _{2} \int _{\Sigma (\lambda ,t)} \rho _{2}^{2} \,dS \\ &\qquad{}+\gamma \int _{\partial \omega (\lambda +t)} \rho _{2}\biggl( \frac{{\mathbf {v}}}{2}- \kappa _{2}\nabla _{\Sigma }\rho _{2}\biggr) \cdot { \mathbf{e}}_{3} \,dl- \gamma \int _{\partial \omega (\lambda -t)} \rho _{2}\biggl( \frac{{\mathbf {v}}}{2}- \kappa _{2}\nabla _{\Sigma }\rho _{2}\biggr) \cdot { \mathbf{e}}_{3}\,dl \\ &\qquad{}- \int _{\omega (\lambda +t)}{\mathbf{T}}({\mathbf {v}},p)\mathbf{e}_{3} \cdot { \mathbf {v}} \,dx'+ \int _{\omega (\lambda -t)}{\mathbf{T}}({\mathbf {v}},p){ \mathbf{e}}_{3} \cdot {\mathbf {v}} \,dx' \\ &\quad= \int _{\Omega (\lambda ,t)} {\mathbf{f}}\cdot {\mathbf {v}} \,dx+ \int _{ \Sigma (\lambda ,t)}{\mathbf{b}}\cdot {\mathbf {v}} \,dS, \end{aligned}$$
(57)
where \(\mathbf{e}_{3}=(0,0,1)\) and \(\omega (t)=\{x_{3}=t, x'\in B\}\).
We integrate (57) with respect to t over the interval \((\eta -1,\eta ) (1<\eta )\) to obtain
$$\begin{aligned} Z(\lambda ,\eta ) \equiv{}& \int ^{\eta }_{\eta -1} \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2, \Omega (\lambda ,t)} \bigr)^{2}+ \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2, \Sigma (\lambda ,t)} \bigr)^{2} \,dt \\ \leq{}& C \biggl\{ \biggl\vert \int _{\tilde{D}(\lambda ,\eta )}{\mathbf{T}}({ \mathbf {v}},p)\mathbf{e}_{3} \cdot {\mathbf {v}} \,dx \biggr\vert + \biggl\vert \int _{D( \lambda ,\eta )}{\mathbf{T}}({\mathbf {v}},p)\mathbf{e}_{3} \cdot {\mathbf {v}} \,dx \biggr\vert \\ &{} + \biggl\vert \int _{\tilde{S}(\lambda ,\eta )} \rho _{2}\biggl( \frac{1}{2}{ \mathbf {v}}-\nabla _{\Sigma }\rho _{2}\biggr) \cdot { \mathbf{e}}_{3} \,dS \biggr\vert + \biggl\vert \int _{S(\lambda ,\eta )} \rho _{2}\biggl(\frac{1}{2}{ \mathbf {v}}-\nabla _{\Sigma }\rho _{2}\biggr) \cdot { \mathbf{e}}_{3} \,dS \biggr\vert \\ &{} + \biggl\vert \int ^{\eta }_{\eta -1} \biggl( \int _{\Omega ( \lambda ,t)} {\mathbf{f}}\cdot {\mathbf {v}} \,dx \biggr)\,dt \biggr\vert + \biggl\vert \int ^{\eta }_{\eta -1} \biggl( \int _{\Sigma (\lambda ,t)} {\mathbf{b}}\cdot { \mathbf {v}} \,dS \biggr)\,dt \biggr\vert \biggr\} , \end{aligned}$$
(58)
where
$$\begin{aligned} \textstyle\begin{cases} \tilde{D}(\lambda ,\eta )= \{ x\in \Omega | \lambda +\eta -1< x_{3}< \lambda +\eta \} , \\ D(\lambda ,\eta )= \{ x\in \Omega | \lambda -\eta < x_{3}< \lambda -\eta +1 \} , \\ \tilde{S}(\lambda ,\eta )= \{ x\in \Sigma | \lambda +\eta -1< x_{3}< \lambda +\eta \} , \\ S(\lambda ,\eta )= \{ x\in \Sigma | \lambda -\eta < x_{3}< \lambda -\eta +1 \} . \end{cases}\displaystyle \end{aligned}$$
The terms on the right-hand side are estimated as follows:
$$\begin{aligned} &\biggl| \int _{\tilde{D}(\lambda ,\eta )}{\mathbf{T}}({\mathbf {v}},p)\mathbf{e}_{3} \cdot {\mathbf {v}} \,dx\biggr|\leq C \bigl\{ \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2, \tilde{D}(\lambda ,\eta )} \bigr)^{2}+ \Vert {\mathbf{f}} \Vert _{2,\tilde{D}( \lambda ,\eta )}^{2} \bigr\} , \\ &\biggl| \int _{D(\lambda ,\eta )}{\mathbf{T}}({\mathbf {v}},p)\mathbf{e}_{3} \cdot {\mathbf {v}} \,dx\biggr|\leq C \bigl\{ \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2,D( \lambda ,\eta )} \bigr)^{2}+ \Vert {\mathbf{f}} \Vert _{2,D(\lambda ,\eta )}^{2} \bigr\} , \\ &\biggl\vert \int _{\tilde{S}(\lambda ,\eta )} \rho _{2}\biggl(\frac{1}{2}{ \mathbf {v}}-\nabla _{\Sigma }\rho _{2}\biggr) \cdot { \mathbf{e}}_{3} \,dS \biggr\vert \leq C \bigl\{ \bigl( \Vert { \mathbf {v}} \Vert ^{(1)}_{2,\tilde{D}(\lambda , \eta )} \bigr)^{2}+ \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,\tilde{S}(\lambda , \eta )} \bigr)^{2} \bigr\} , \\ &\biggl\vert \int _{S(\lambda ,\eta )} \rho _{2}\biggl(\frac{1}{2}{ \mathbf {v}}- \nabla _{\Sigma }\rho _{2}\biggr) \cdot { \mathbf{e}}_{3} \,dS \biggr\vert \leq C \bigl\{ \bigl( \Vert { \mathbf {v}} \Vert ^{(1)}_{2,D(\lambda ,\eta )} \bigr)^{2}+ \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,S(\lambda ,\eta )} \bigr)^{2} \bigr\} , \end{aligned}$$
and
$$\begin{aligned} & \biggl\vert \int ^{\eta }_{\eta -1} \biggl( \int _{\Omega (\lambda ,t)} {\mathbf{f}} \cdot {\mathbf {v}} \,dx \biggr)\,dt \biggr\vert + \biggl\vert \int ^{\eta }_{\eta -1} \biggl( \int _{\Sigma (\lambda ,t)} {\mathbf{b}}\cdot {\mathbf {v}} \,dS \biggr)\,dt \biggr\vert \\ &\quad\leq \epsilon Z(\lambda ,\eta )+C(\epsilon ) \bigl( \Vert {\mathbf{f}} \Vert _{2, \Omega (\lambda ,\eta )}^{2}+ \Vert {\mathbf{b}} \Vert _{2,\Sigma (\lambda ,\eta )}^{2} \bigr), \end{aligned}$$
where \(\epsilon >0\) is an arbitrary constant.
Thus, from (58), we have
$$\begin{aligned} Z(\lambda ,\eta )\leq{}& C_{1} \bigl\{ \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2, \tilde{D}(\lambda ,\eta )} \bigr)^{2}+ \bigl( \Vert { \mathbf {v}} \Vert ^{(1)}_{2,D( \lambda ,\eta )} \bigr)^{2} \\ &{}+ \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,\tilde{S}(\lambda ,\eta )} \bigr)^{2} + \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,S(\lambda ,\eta )} \bigr)^{2} \bigr\} +C_{2} \bigl( \Vert {\mathbf{f}} \Vert _{2,\Omega (\lambda ,\eta )}^{2}+ \Vert { \mathbf{b}} \Vert _{2,\Sigma (\lambda ,\eta )}^{2} \bigr) \\ ={}&C_{1}\frac{d}{d\eta }Z(\lambda ,\eta )+C_{2} \bigl( \Vert {\mathbf{f}} \Vert _{2, \Omega (\lambda ,\eta )}^{2}+ \Vert {\mathbf{b}} \Vert _{2,\Sigma (\lambda ,\eta )}^{2} \bigr). \end{aligned}$$
Multiplying this inequality by \(e^{-\eta /c_{1}}\) and integrating over the interval \((2,-\lambda /2)\) with respect to η gives
$$\begin{aligned} &Z(\lambda ,2) \\ &\quad\leq Z\biggl(\lambda ,-\frac{\lambda }{2}\biggr)e^{\frac{\lambda +4}{2C_{1}}} + \frac{C_{2}}{C_{1}} \int ^{-\frac{\lambda }{2}}_{2} \bigl( \Vert {\mathbf{f}} \Vert _{2, \Omega (\lambda ,\eta )}^{2} + \Vert {\mathbf{b}} \Vert _{2,\Sigma (\lambda ,\eta )}^{2} \bigr)e^{-\frac{\eta -2}{C_{1}}}\,d\eta . \end{aligned}$$
(59)
Now, we note the following inequalities:
$$\begin{aligned} &\bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2,\Omega (\lambda ,1)} \bigr)^{2} + \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,\Sigma (\lambda ,1)} \bigr)^{2}\leq Z( \lambda ,2), \\ &Z\biggl(\lambda ,-\frac{\lambda }{2}\biggr)\leq \bigl( \Vert {\mathbf {v}} \Vert ^{(1)}_{2, \Omega } \bigr)^{2} + \bigl( \Vert \rho _{2} \Vert ^{(1)}_{2,\Sigma } \bigr)^{2} \\ &\phantom{Z\biggl(\lambda ,-\frac{\lambda }{2}\biggr)} \leq C \bigl\{ \bigl( \vert {\mathbf{f}} \vert ^{(\alpha ,\mu )}_{s-2,\Omega ,M} \bigr)^{2}+\bigl( \vert {\mathbf{a}} \vert _{s-1,\Gamma ,M}^{(1+\alpha )} \bigr)^{2}+\bigl( \vert {\mathbf{b}} \vert ^{(1+ \alpha ,\mu )}_{s-1,\Sigma ,M} \bigr)^{2} \bigr\} , \end{aligned}$$
and
$$\begin{aligned} & \int ^{-\frac{\lambda }{2}}_{2} \bigl( \Vert {\mathbf{f}} \Vert _{2,\Omega (\lambda , \eta )}^{2}+ \Vert {\mathbf{b}} \Vert _{2,\Sigma (\lambda ,\eta )}^{2} \bigr)e^{- \frac{\eta -2}{C_{1}}}\,d\eta \\ &\quad \leq Ce^{2\mu \lambda } \bigl\{ \bigl( \vert {\mathbf{f}} \vert ^{(\alpha ,\mu )}_{s-2, \Omega ,M}\bigr)^{2}+\bigl( \vert {\mathbf{b}} \vert ^{(1+\alpha ,\mu )}_{s-1,\Sigma ,M}\bigr)^{2} \bigr\} \end{aligned}$$
to obtain the desired estimate (56) for \(0<\mu <1/2C_{1}\). □
Let us proceed to the estimation of the higher-order derivatives of the solution through Schauder’s method. We introduce a covering \(\{U_{i}\}_{i\in \mathbb{N}}\) of Ω̄, where \(\{U_{i}\}\) is a family of balls with the following properties:
(i) if \(U_{i}\cap M \neq \phi \), then \(\xi _{i}\), the center of \(U_{i}\), is on M; if \(U_{i}\cap M = \phi \) and \(U_{i}\cap \partial \Omega \neq \phi \), then \(\xi _{i}\) is on Γ or Σ;
(ii) the radius of \(U_{i}\) (\(U_{i}\cap \partial \Omega \neq \phi \)) is sufficiently small that \(U_{i}\) divides the surface ∂Ω into two connected parts; and
(iii) there exists an integer \(N_{0}\) such that the intersection of \(N_{0}+1\) arbitrary different balls is empty.
We further introduce a partition of unity \(\{\zeta _{i}\}\) corresponding to the covering \(\{U_{i}\}\).
For \(U_{i}\) with \(\xi _{i} \notin M\), the following estimates are obtained (for the derivation, see [11]):
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha )}_{\Omega '_{i}}+ \vert \nabla p \vert ^{(\alpha )}_{ \Omega '_{i}}\leq C \bigl( \vert { \mathbf{f}} \vert ^{(\alpha )}_{\Omega _{i}}+ \Vert { \mathbf {v}} \Vert _{2,\Omega _{i}} \bigr)\quad (\xi _{i} \in \Omega ), \end{aligned}$$
(60)
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha )}_{\Omega _{i}'} + \vert \nabla p \vert ^{(\alpha )}_{ \Omega _{i}'}+ \vert \rho _{1} \vert ^{(3+\alpha )}_{\Gamma _{i}'} \\ & \quad\leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha )}_{\Omega _{i}}+ \vert {\mathbf{a}} \vert ^{(1+ \alpha )}_{\Gamma _{i}} + \Vert {\mathbf {v}} \Vert _{2,\Omega _{i}}+ \Vert \rho _{1} \Vert _{2,\Gamma _{i}} \bigr)\quad (\xi _{i} \in \Gamma ), \end{aligned}$$
(61)
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha )}_{\Omega _{i}'} + \vert \nabla p \vert ^{(\alpha )}_{ \Omega _{i}'}+ \vert \rho _{2} \vert ^{(3+\alpha )}_{\Sigma _{i}'} \\ &\quad \leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha )}_{\Omega _{i}}+ \vert {\mathbf{b}} \vert ^{(1+ \alpha )}_{\Sigma _{i}} + \Vert {\mathbf {v}} \Vert _{2,\Omega _{i}}+ \Vert \rho _{2} \Vert _{2,\Sigma _{i}} \bigr) \quad(\xi _{i} \in \Sigma ), \end{aligned}$$
(62)
where \(D_{i}\) and \(D'_{i}\) denote the domains \(D\cap U_{i}\) and \(D\cap U'_{i}\), respectively, for \(D=\Omega , \Gamma , \Sigma \), and \(U'_{i}\) denotes a subset of \(U_{i}\) on which \(\zeta _{i}=1\) holds. To estimate the solution defined on \(U_{i}\) with \(\xi _{i} \in M\), we use the axial symmetry of the solution and rewrite problem (32) as the following two-dimensional problem in cylindrical coordinates:
$$\begin{aligned} \textstyle\begin{cases} {-}\nu \Delta {\mathbf {v}}+\nabla p={\mathbf {\mathcal {F}}}, \\ \partial _{r} v_{r} + \partial _{z} v_{z}=-\frac{v_{r}}{r}\equiv { \mathcal{G}} \quad \text{in } \Omega , \\ (2\nu {\mathbf{D}}({\mathbf {v}})\mathbf{n}+\gamma \nabla _{\Gamma }\rho _{1} )\cdot {\mathbf {\tau }}=\mathbf{a}\cdot {\mathbf {\tau }}, \qquad { \mathbf {v}}\cdot {\mathbf{n}}=0 \\ {-}\frac{\kappa _{1}}{r\sqrt{1+(h')^{2}}}\frac{d}{dr}( \frac{r}{\sqrt{1+(h')^{2}}}\frac{d \rho _{1}}{dr} ) \\ \quad{} +\chi _{1}\rho _{1}+\nabla _{\Gamma }\cdot {\mathbf {v}}=0 \quad\text{on } \Gamma , \\ (2\nu {\mathbf{D}}({\mathbf {v}})\mathbf{n}+\beta {\mathbf {v}}+ \frac{\gamma }{2} \nabla _{\Sigma }\rho _{2} )\cdot {\mathbf {\tau }}={ \mathbf{b}}\cdot {\mathbf {\tau }}, \qquad {\mathbf {v}}\cdot {\mathbf{n}}=0 \\ {-}\kappa _{2}\frac{d^{2}\rho _{2}}{dz^{2}}+\chi _{2}\rho _{2}+ \frac{1}{2}\nabla _{\Sigma }\cdot {\mathbf {v}}=0 \quad \text{on } \Sigma , \\ \frac{\kappa _{1}}{\sqrt{1+h'(1)^{2}}}\frac{d\rho _{1}}{d r}+{ \mathbf {v}}\cdot {\mathbf{e}}_{1}=0, \\ \kappa _{2}\frac{d\rho _{2}}{dz}+\frac{1}{2}{\mathbf {v}}\cdot {\mathbf{e}}_{2}=0\quad \text{at } M, \end{cases}\displaystyle \end{aligned}$$
(63)
where
$$\begin{aligned} \textstyle\begin{cases} \Delta =\partial _{r}^{2}+\partial _{z}^{2}, \qquad \nabla = \binom{\partial _{r}}{\partial _{z}}, \qquad {\mathbf {\mathcal{F}}}=(\nu (\frac{1}{r}\partial _{r}v_{r}-\frac{v_{r}}{r^{2}} )+f_{r}, \frac{\nu }{r}\partial _{r} v_{z}+f_{z}), \\ \mathbf{D}({\mathbf {v}})=\frac{1}{2} \begin{pmatrix} 2\partial _{r}v_{r}&\partial _{z}v_{r}+\partial _{r}v_{z} \\ \partial _{z}v_{r}+\partial _{r}v_{z}&2\partial _{z}v_{z} \end{pmatrix} , \\ \Omega = \{ z< h(r), 0\leq r< 1 \} ,\qquad \Gamma = \{ z=h(r), 0\leq r< 1 \} , \\ \Sigma = \{ r=1, z< 0 \} ,\qquad M=\bar{\Gamma }\cap \bar{\Sigma }=(1,0), \\ \mathbf{n}=(n_{r},n_{z})=\textstyle\begin{cases} (-\frac{h'}{\sqrt{1+(h')^{2}}},\frac{1}{\sqrt{1+(h')^{2}}} )\quad \text{on } \Gamma , \\ (1,0)\quad \text{on } \Sigma , \end{cases}\displaystyle \\ {\mathbf {\tau }}=(\tau _{r},\tau _{z})=(n_{z},-n_{r}), \\ \mathbf{e}_{1}= \biggl(-\frac{1}{\sqrt{1+h'(1)^{2}}},- \frac{h'(1)}{\sqrt{1+h'(1)^{2}}} \biggr), \qquad\mathbf{e}_{2}=(0,-1), \end{cases}\displaystyle \end{aligned}$$
(64)
and \(f_{r}\) and \(f_{z}\) denote the r-component and z-component, respectively, of the term f.
We rewrite the above problem in the coordinates \((y_{1},y_{2})\), which are related to the original coordinates by \(\binom{y_{1}}{y_{2}}=\mathcal{R}\binom{r-1}{z}\), where \(\mathcal{R}\) is the matrix of rotation \(\theta -(3/2)\pi \) around the origin. In the coordinates \(\{y\}\), Σ is given by the line \(y_{2}=y_{1}\tan \theta \) (or \(y_{1}=0\) when \(\theta =\pi /2\)), and we represent Γ as the curve \(y_{2}=g(y_{1})\).
We now introduce a mapping \(y=\Phi (\eta )=(\eta _{1},\eta _{2})+(\Phi _{1}(\eta ),\Phi _{2}( \eta ))\), where \(\Phi _{1}\) and \(\Phi _{2}\) are constructed to satisfy the following conditions:
$$\begin{aligned} \bigl(\Phi _{1}(\eta ),\Phi _{2}(\eta )\bigr)|_{\gamma _{0}}= \bigl(0,\tilde{g}( \eta _{1}) \bigr),\qquad \bigl(\Phi _{1}(\eta ), \Phi _{2}(\eta )\bigr)|_{ \gamma _{\theta }}=(0,0), \end{aligned}$$
in which g̃ is an extension of g on the half-line \({\mathbb{R}}_{+}\) in the class \(C^{3+\alpha }_{s+1}({\mathbb{R}}_{+},0)\).
Then, the functions \({\mathbf {u}}=(u_{1},u_{2})=\hat{\zeta }\hat{\mathbf {v}}\equiv (\zeta { \mathbf {v}})\circ \Phi \), \(q=\hat{\zeta }\hat{p}\equiv (\zeta p)\circ \Phi \), \(r_{1}=\hat{\zeta }\hat{\rho }_{1}\equiv (\zeta \rho _{1})\circ \Phi |_{ \gamma _{0}}\), and \(r_{2}=\hat{\zeta }\hat{\rho }_{2}\equiv (\zeta \rho _{2})\circ \Phi |_{ \gamma _{\theta }}\) (here the same symbols v, p, \(\rho _{1}\), and \(\rho _{2}\) are used to denote the unknowns defined in the coordinates \(\{y\}\)), where
$$\begin{aligned} \zeta (y)\in C^{\infty }_{0}\bigl(\mathbf{R}^{2} \bigr), \qquad\zeta =1 \biggl( \vert y \vert < \frac{\delta }{2}\biggr), \qquad\zeta =0 \bigl( \vert y \vert >\delta \bigr) \end{aligned}$$
for a constant \(\delta >0\) satisfy the following equations in the sectorial domain \(d_{\theta }\):
$$\begin{aligned} &\textstyle\begin{cases} -\nu \Delta {\mathbf {u}}+\nabla q =\hat{\zeta }\hat{\mathbf {\mathcal{F}}}- \nu (\nabla ^{2} -\hat{\nabla }^{2}){\mathbf {u}}+(\nabla -\hat{\nabla })q+{ \mathbf{F}}, \\ \nabla \cdot {\mathbf {u}}=\hat{\zeta }\hat{\mathcal{G}}+(\nabla - \hat{\nabla })\cdot {\mathbf {u}}+G \quad \text{in } \,d_{\theta }, \end{cases}\displaystyle \end{aligned}$$
(65)
$$\begin{aligned} &\textstyle\begin{cases} 2\nu {\mathbf{D}}({\mathbf {u}})\mathbf{n}\cdot {\mathbf {\tau }} =\hat{\zeta }( \hat{\mathbf{a}}-\gamma \hat{\nabla }_{\Gamma }\hat{\rho _{1}})\cdot \hat{\mathbf {\tau }} \\ \quad{} +2\nu \{ \mathbf{D}({\mathbf {u}})\mathbf{n}\cdot {\mathbf {\tau }}- \hat{\mathbf{D}}({\mathbf {u}})\hat{\mathbf{n}}\cdot \hat{\mathbf {\tau }} \} +A, \\ {\mathbf {u}}\cdot {\mathbf{n}}={\mathbf {u}}\cdot (\mathbf{n}-\hat{\mathbf{n}}), \\ {-\kappa _{1}\frac{d^{2}r_{1}}{d\eta _{1}^{2}}+\chi _{1}r_{1}=- \hat{\zeta }\hat{\nabla }_{\Gamma }\cdot \hat{\mathbf {v}}} \\ \quad{} -\kappa _{1}((\frac{d}{d\eta _{1}})^{2}-( \hat{\frac{d}{d\eta _{1}}})^{2})r_{1}+H_{1} \quad \text{on } \gamma _{0}, \end{cases}\displaystyle \end{aligned}$$
(66)
$$\begin{aligned} & \textstyle\begin{cases} {2\nu {\mathbf{D}}({\mathbf {u}})\mathbf{n}\cdot {\mathbf {\tau }}= \hat{\zeta }(\hat{\mathbf{b }}-\beta \hat{\mathbf {v}}-\frac{\gamma }{2} \hat{\nabla }_{\Sigma }\hat{\rho _{2}})\cdot \hat{\mathbf {\tau }}} \\ \quad{} +2\nu \{ \mathbf{D}({\mathbf {u}})\mathbf{n}\cdot {\mathbf {\tau }}- \hat{\mathbf{D}}({\mathbf {u}})\hat{\mathbf{n}}\cdot \hat{\mathbf {\tau }} \} +B, \\ {\mathbf {u}}\cdot {\mathbf{n}}={\mathbf {u}}\cdot (\mathbf{n}-\hat{\mathbf{n}}), \\ {-\kappa _{2}\cos ^{2}\theta \frac{d^{2}r_{2}}{d\eta _{1}^{2}}+\chi _{2}r_{2}=-\frac{1}{2} \hat{\zeta } \hat{\nabla }_{\Sigma }\cdot \hat{\mathbf {v}}} \\ \quad{} -\kappa _{2}\cos ^{2}\theta (( \frac{d}{d\eta _{1}})^{2}-(\hat{\frac{d}{d\eta _{1}}})^{2})r_{2}+H_{2} \quad\text{on } \gamma _{\theta }, \end{cases}\displaystyle \end{aligned}$$
(67)
$$\begin{aligned} & \textstyle\begin{cases} {-\kappa _{1}\frac{dr_{1}}{d\eta _{1}}=-\hat{\zeta } \hat{\mathbf {v}}\cdot \hat{\mathbf{e}}_{1}-\kappa _{1}(\frac{d}{d\eta _{1}}- \hat{\frac{d}{d\eta _{1}}})r_{1}-\kappa _{1}\hat{\rho }_{1}( \hat{\frac{d}{d\eta _{1}}})\hat{\zeta }}, \\ {-\kappa _{2}\cos \theta \frac{dr_{2}}{d\eta _{1}}=- \frac{1}{2}\hat{\zeta } \hat{\mathbf {v}}\cdot \hat{\mathbf{e}}_{2}-\kappa _{2} \cos \theta (\frac{d}{d\eta _{1}}-\hat{\frac{d}{d\eta _{1}}})r_{2}} \\ \quad{} {-\kappa _{2}\cos \theta \hat{\rho }_{2}( \hat{\frac{d}{d\eta _{1}}})\hat{\zeta }}\quad \text{on } M\equiv \bar{\gamma }_{0}\cap \bar{\gamma }_{\theta }, \end{cases}\displaystyle \end{aligned}$$
(68)
where
$$\begin{aligned} \textstyle\begin{cases} {\mathbf{F}}=-\nu \{ 2(\hat{\nabla } \hat{\zeta } \cdot \hat{\nabla }) \hat{\mathbf {v}}+\hat{\mathbf {v}}\hat{\nabla }^{2}\hat{\zeta } \} +( \hat{\nabla } \hat{\zeta })\hat{p},\qquad G=\hat{\nabla } \hat{\zeta } \cdot \hat{\mathbf {v}}, \\ A=B=\nu \{ \hat{\nabla }\hat{\zeta } (\hat{\mathbf {v}}\cdot \hat{\mathbf{n}})+\hat{\mathbf {v}}(\hat{\nabla } \hat{\zeta }\cdot \hat{\mathbf{n}})-2(\hat{\mathbf {v}}\cdot \hat{\mathbf{n}})(\hat{\nabla } \hat{\zeta }\cdot \hat{\mathbf{n}})\hat{\mathbf{n}} \} , \\ {H_{1}=-\kappa _{1}\{(\hat{\frac{d}{d\eta _{1}}})^{2}r_{1}- \hat{\zeta }{\mathcal{G}}_{1}\hat{\frac{d}{d\eta _{1}}}(\mathcal{G}_{2} \hat{\frac{d}{d\eta _{1}}}\hat{\rho }_{1})\}}, \\ {H_{2}=-\kappa _{2}\cos ^{2}\theta \{( \hat{\frac{d}{d\eta _{1}}})^{2}r_{2}-\hat{\zeta }( \hat{\frac{d}{d\eta _{1}}})^{2}\hat{\rho }_{2}\}}. \end{cases}\displaystyle \end{aligned}$$
(69)
When \(\theta =\pi /2\), equations (67)3 and (68)2 are replaced by \(-\kappa _{2}\frac{d^{2}r_{2}}{d\eta _{2}^{2}}+\chi _{2}r_{2}=- \frac{1}{2}\hat{\zeta } \hat{\nabla }_{\Sigma }\cdot \hat{\mathbf {v}}\) and \(-\kappa _{2}\frac{dr_{2}}{d\eta _{2}}=-\frac{1}{2}\hat{\zeta } \hat{\mathbf {v}}\cdot \hat{\mathbf{e}}_{2}\), respectively.
Here, \(\hat{\nabla }=({}^{t}{\mathcal{A}}^{-1})\nabla \), where \(\mathcal{A}\) is the Jacobian matrix of the mapping Φ and tA denotes the transpose of matrix A; \(\hat{\mathbf{D}}(\hat{\mathbf {v}})\) denotes the tensor with elements \(\frac{1}{2}\sum_{k=1}^{3} (a_{ik}\partial _{\eta _{k}} \hat{v}_{j}+a_{jk}\partial _{\eta _{k}} \hat{v}_{i} )\ (i,j=1,2) \), where \(a_{ij}\) is the \((i,j)\)th element of \({}^{t}{\mathcal{A}}^{-1}\); \(\hat{\nabla }_{\Gamma }\) and \(\hat{\nabla }_{\Sigma }\) are defined by \(\hat{\nabla }-(\hat{\mathbf{n}}\cdot \hat{\nabla })\hat{\mathbf{n}}\); \(\hat{\frac{d}{d\eta _{1}}}=\hat{\partial }_{\eta _{1}}+ \frac{d\hat{\tilde{g}}}{d\eta _{1}}\hat{\partial }_{\eta _{2}}\) (\(\binom{\hat{\partial }_{\eta _{1}}}{\hat{\partial }_{\eta _{2}}}= \hat{\nabla }\)) on \(\gamma _{0}\); \(\hat{\frac{d}{d\eta _{1}}}=\hat{\partial }_{\eta _{1}}+\tan \theta \hat{\partial }_{\eta _{2}}\) on \(\gamma _{\theta }\); \(\mathcal{G}_{1}= \frac{1}{(1-y_{1}\sin \theta +y_{2}\cos \theta )\sqrt{1+({\tilde{g}'})^{2}}} \circ \Phi \); and \(\mathcal{G}_{2}= \frac{1-y_{1}\sin \theta +y_{2}\cos \theta }{\sqrt{1+({\tilde{g}'})^{2}}} \circ \Phi \).
By applying estimate (27) to the system consisting of (65), (66)1,2, and (67)1,2 and applying estimate (44) to the system consisting of (66)3, \((\text{67})^{3}\), and (68), we have
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha )}_{s,\Omega '_{i} ,M'_{i}} + \vert \nabla p \vert ^{( \alpha )}_{s-2,\Omega '_{i},M'_{i}}+ \vert \rho _{1} \vert ^{(3+\alpha )}_{s+1, \Gamma '_{i},M'_{i}} + \vert \rho _{2} \vert ^{(3+\alpha )}_{s+1,\Sigma '_{i},M'_{i}} \\ &\quad\leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha )}_{s-2,\Omega _{i},M_{i}} + \vert {\mathbf{a}} \vert ^{(1+ \alpha )}_{s-1,\Gamma _{i},M_{i}} + \vert { \mathbf{b}} \vert ^{(1+\alpha )}_{s-1, \Sigma _{i},M_{i}} \\ &\qquad{} + \Vert {\mathbf {v}} \Vert _{2,\Omega _{i}} + \Vert \rho _{1} \Vert _{2, \Gamma _{i}}+ \Vert \rho _{2} \Vert _{2,\Sigma _{i}} \bigr). \end{aligned}$$
(70)
Let \(\{U_{i}\}_{i\in I}\) be a finite covering of \(\bar{\Omega }_{0,4}\) which is chosen from the set \(\{U_{i}\}_{i\in \mathbb{N}}\). If we assume that each element of \(\{U_{i}\}_{i\in I_{1}(\subset I)}\) satisfies \(U_{i}\cap M=\phi \), then the norms \(|\cdot |^{(l+\alpha )}_{s,\Omega _{i},M}\) and \(|\cdot |^{(l+\alpha )}_{\Omega _{i}}\) are equivalent for \(i\in I_{1}\), since any point in \(\Omega _{i}\ (i\in I_{1})\) is away from M by a positive distance. Thus, by summing estimates (60)–(62) and (70) for all \(i\in I\) and then using (55), we can obtain the estimate
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha )}_{s,\Omega _{0,4},M} + \vert \nabla p \vert ^{( \alpha )}_{s-2,\Omega _{0,4},M}+ \vert \rho _{1} \vert ^{(3+\alpha )}_{s+1, \Gamma ,M} + \vert \rho _{2} \vert ^{(3+\alpha )}_{s+1,\Sigma _{0,4},M} \\ &\quad\leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha ,\mu )}_{s-2,\Omega ,M}+ \vert {\mathbf{a}} \vert ^{(1+ \alpha )}_{s-1,\Gamma ,M} + \vert { \mathbf{b}} \vert ^{(1+\alpha ,\mu )}_{s-1,\Sigma ,M} \bigr). \end{aligned}$$
(71)
In a similar manner, we take estimates (60) and (62) and use (56) in the domain \(\Omega (\lambda ,1-\delta )\) to obtain the estimate
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha )}_{\Omega (\lambda ,1-\delta )}+ \vert \nabla p \vert ^{( \alpha )}_{\Omega (\lambda ,1-\delta )}+ \vert \rho _{2} \vert ^{(3+\alpha )}_{ \Sigma (\lambda ,1-\delta )} \\ &\quad\leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha )}_{\Omega (\lambda ,1)}+ \vert {\mathbf{b}} \vert ^{(1+ \alpha )}_{\Sigma (\lambda ,1)}+ \Vert {\mathbf {v}} \Vert _{2,\Omega (\lambda ,1)}+ \Vert \rho _{2} \Vert _{2,\Sigma (\lambda ,1)} \bigr) \\ &\quad\leq C \bigl\{ \vert {\mathbf{f}} \vert ^{(\alpha )}_{\Omega (\lambda ,1)}+ \vert {\mathbf{b}} \vert ^{(1+ \alpha )}_{\Sigma (\lambda ,1)} \\ &\qquad{} +e^{\mu \lambda }\bigl( \vert {\mathbf{f}} \vert ^{(\alpha , \mu )}_{s-2,\Omega ,M}+ \vert {\mathbf{a}} \vert ^{(1+\alpha )}_{s-1,\Gamma ,M}+ \vert {\mathbf{b}} \vert ^{(1+ \alpha ,\mu )}_{s-1,\Sigma ,M}\bigr) \bigr\} , \end{aligned}$$
(72)
where \(0<\delta <1\) is a constant. Multiplying both sides of (72) by \(e^{-\mu \lambda }\) and taking the supremum for \(\lambda <-4\), and then noting that the norms \(|e^{-\mu x_{3}}{\mathbf {v}}|^{(2+\alpha )} _{\Omega _{3+\delta }}\) and \(\sup_{\lambda <-4}e^{-\mu \lambda }|{\mathbf {v}}|^{(2+\alpha )}_{ \Omega (\lambda ,1-\delta )}\) are equivalent, we have
$$\begin{aligned} & \vert {\mathbf {v}} \vert ^{(2+\alpha ,\mu )}_{\Omega _{3+\delta }}+ \vert \nabla p \vert ^{( \alpha ,\mu )}_{\Omega _{3+\delta }}+ \vert \rho _{2} \vert ^{(3+\alpha ,\mu )}_{ \Sigma _{3+\delta }} \\ &\quad\leq C \bigl( \vert {\mathbf{f}} \vert ^{(\alpha ,\mu )}_{s-2,\Omega ,M}+ \vert {\mathbf{a}} \vert ^{(1+ \alpha )}_{s-1,\Gamma ,M}+ \vert {\mathbf{b}} \vert ^{(1+\alpha ,\mu )}_{s-1,\Sigma ,M} \bigr). \end{aligned}$$
(73)
Thus, from (71) and (73), we reach the desired result.