1 Introduction

Let \({\mathcal {E}}\) be an exterior domain in \({{\mathbb {R}}}^2\), that is, \({\mathcal {E}}={{\mathbb {R}}}^2\setminus \overline{{\mathcal {B}}}\), where \({\mathcal {B}}\) is a bounded open set with sufficiently smooth boundary.Footnote 1 To be definite, we assume that the origin is \(0 \in {\mathcal {B}}\). Without loss of generality, we also let \({\mathbb {R}}^2 \setminus B_1 \subset {\mathcal {E}}\), where \(B_1\) is the open unit disk centered at 0. This paper studies the stationary Navier–Stokes equations in \({\mathcal {E}}\) with a Dirichlet boundary condition on \(\partial {\mathcal {E}}\) and nonzero prescribed velocity at spatial infinity, that is,

figure a

The parameter \(\lambda > 0\) will be referred to as the Reynolds number. Here \(\mathbf{e }_1=(1,0)\) is the unit vector along x-axis. Physically, the system (\(1.1_\lambda \)) describes the stationary motion of a viscous incompressible fluid flowing past a rigid cylindrical body.

This problem has its origins in the 19th century, starting with the classical paper of Stokes [31], where the famous paradox of his name was discovered, that is, that the corresponding linearized system

$$\begin{aligned} \left\{ \begin{aligned}&\Delta \mathbf{w } - \nabla p = 0, \\&\nabla \cdot \mathbf{w } = 0, \\&\mathbf{w }|_{\partial {\mathcal {E}}} = 0, \\&\mathbf{w }(z) \rightarrow \mathbf{e }_1 \ \ \text {as}\ \ |z| \rightarrow \infty \end{aligned} \right. \end{aligned}$$
(1.1)

has no solution.Footnote 2 The mathematical nature of the Stokes paradox was the subject of many investigations, see, for example, [4, 26].

The celebrated J.Leray’s paper [23] can be considered as a landmark point in the study of the nonlinear problem (\(1.1_\lambda \)). There, among many other results, Leray suggested an elegant approach which was called “the invading domains method ”. Denoting by \({\mathbf{w }}_k\) the solution to the problem

$$\begin{aligned} \left\{ \begin{array}{ll} - \Delta {{\mathbf{w }}_k}+({\mathbf{w }}_k\cdot \nabla ){\mathbf{w }}_k+ \nabla p_k {} =\mathbf{0} &{}\qquad \hbox {in } {{\mathcal {E}}}\cap B_{R_k}, \\ \hbox {div}\,{{\mathbf{w }}_k} =0 &{}\qquad \hbox {in } {{\mathcal {E}}}\cap B_{R_k}, \\ {{\mathbf{w }}_k} = \mathbf{0 } &{}\qquad \hbox {on } \partial {{\mathcal {E}}}, \\ {\mathbf{w }}_k ={\mathbf{w }}_\infty &{}\quad \text{ for } |z|=R_k \end{array}\right. \end{aligned}$$
(1.2)

on the intersection of \({{\mathcal {E}}}\) with the disk \(B_{R_k}\) of radius \(R_k{\,\geqq \,}k(\gg R_0)\), whose existence he proved before, Leray showed that the sequence \({\mathbf{w }}_k\) satisfies the estimate

$$\begin{aligned} \int _{{{\mathcal {E}}}\cap B_{R_k}}|\nabla {\mathbf{w }}_k|^2{\,\leqq \,}c \end{aligned}$$
(1.3)

for some positive constant c independent of k. Hence, he observed that it is possible to extract a subsequence \({\mathbf{w }}_{k_n}\) which weakly converges to a solution \(\mathbf{w }_L\) of problem (\(1.1_\lambda \))\({}_{1,2,3}\) with \(\int _{{\mathcal {E}}}|\nabla {\mathbf{w }}_L|^2<+\infty \).Footnote 3 This solution was later called Leray’s solution (see, e.g., [1] ).

This achievement of Leray immediately raises two crucial questions:

(1) Is the constructed solution \({\mathbf{w }}_L\) nontrivial, that is, can we exclude the identity \({\mathbf{w }}_L\equiv \mathbf{0 }\)?

This question is rather natural, since if we apply the Leray “invading domains” method to the corresponding Stokes system (1.1) (or even to the simplest Laplace equation), then the limiting solution will be identically zero.

(2) If \({\mathbf{w }}_L\) is nontrivial, what can we say about its behavior at infinity? Namely, can we guarantee the desired convergence

$$\begin{aligned} \mathbf{w }_{L}(z) \rightarrow {\mathbf{w }}_\infty \ \ \text {as}\ \ |z| \rightarrow \infty \ \ \ ? \end{aligned}$$
(1.4)

Many useful properties of Leray solutions were discovered in the classical papers by D. Gilbarg and H.F. Weinberger [11, 12]. Further more, in the very deep paper [1] Ch. Amick proved, under an additional axial symmetry assumption, that the Leray solutions are nontrivial and they have some uniform limits at infinity, that is, there exists a constant vector \(\mathbf{w }_0\in {{\mathbb {R}}}^2\) such that

$$\begin{aligned} \mathbf{w }_L(z) \rightarrow \mathbf{w }_0 \ \ \text {as}\ \ |z| \rightarrow \infty . \end{aligned}$$
(1.5)

Very recently, in the joint papers by Korobkov–Pileckas–Russo [18,19,20], this additional symmetry assumption was removed, that is, they proved that Leray solutions are always nontrivial and have some uniform limit at infinity (1.5).

Nevertheless, despite the classical papers and the recent progress, the fundamental question, whether or not the Leray solutions satisfy the limiting condition (1.4), that is, whether the equality \(\mathbf{w }_0={\mathbf{w }}_\infty \) holds, is still open. In other words, it is not clear whether one can construct the solution to the initial problem (\(1.1_\lambda \)) by Leray’s method.

The brilliant success (for the small Reynolds numbers) was reached in 1967 by another approach. Namely, using the integral representations with the fundamental solution to Oseen linear system (see below (1.12) ) and a contraction mapping argument in some suitable Banach spaces, R. Finn and D.R. Smith proved [7] the following remarkable result:

Theorem 1

(See [7] Corollary 4.2 and Theorem 7.1) There exist constants \(\lambda _0, M_0, {\varepsilon }_0 > 0\) depending only on the geometry of \(\partial {\mathcal {E}}\) such that, for any \(0<\lambda < \lambda _0\), there exists a smooth solution \(\mathbf{w }_{FS}(z; \lambda )\) to (\(1.1_\lambda \)) in \({\mathcal {E}}\) satisfying the pointwise estimate

$$\begin{aligned} |(\mathbf{w }_{FS} - \lambda \mathbf{e }_1)_i(z)| {\,\leqq \,}M_0 |\log \lambda |^{-1} \lambda h_i(\lambda z), \ i=1,2 \end{aligned}$$
(1.6)

for all \(z \in {\mathcal {E}}\). Moreover, if \(\mathbf{w }\) is a smooth solution to (\(1.1_\lambda \)) in \({\mathcal {E}}\) satisfying

$$\begin{aligned} |(\mathbf{w } - \lambda \mathbf{e }_1)_i(z)| {\,\leqq \,}{\varepsilon }_0 \lambda h_i(\lambda z), \ i = 1,2 \end{aligned}$$
(1.7)

for all \(z \in {\mathcal {E}}\), then \(\mathbf{w } \equiv \mathbf{w }_{FS}\).

(In fact, their construction allows nonzero small boundary data \(\mathbf{w } = \mathbf{a }\) on \(\partial {\mathcal {E}}\) with \(|\mathbf{a }-{\mathbf{w }}_\infty |\) small enough, even without zero total flux condition.)

Here the majorant functions \(h_i(\xi )\) are taken as

$$\begin{aligned} 0 < |\xi | {\,\leqq \,}1&: \quad h_i(\xi ) = \log \frac{2}{|\xi |}, \ i = 1,2, \end{aligned}$$
(1.8)
$$\begin{aligned} |\xi |>1&: \quad {\left\{ \begin{array}{ll} h_1(\xi ) = |\xi |^{-\frac{1}{2}}\\ h_2(\xi ) = |\xi |^{-\frac{1}{2}-\mu }, \end{array}\right. } \end{aligned}$$
(1.9)

with \(0<\mu <\frac{1}{2}\) chosen arbitrarily and then fixed. To be definite, we simply take \(\mu = \frac{1}{4}\).

We briefly describe the approach taken by Finn and Smith [7]. (The reader may also consult Galdi [10] Section XII.5 for another approach based on Sobolev norms instead of pointwise bounds.) Let \(\mathbf{v }(z) = \lambda ^{-1} \mathbf{w }(z) - \mathbf{e }_1\). To find a desired solution \(\mathbf{w }(z)\), it is equivalent to finding \(\mathbf{v }\) as a solution to

$$\begin{aligned} \left\{ \begin{aligned}&\Delta \mathbf{v } - \lambda \partial _1 \mathbf{v } - \nabla q = \lambda \mathbf{v } \cdot \nabla \mathbf{v }, \\&\nabla \cdot \mathbf{v } = 0, \\&\mathbf{v }|_{\partial {\mathcal {E}}} = - \mathbf{e }_1, \\&\mathbf{v } \rightarrow 0 \ \ \text {as}\ \ |z| \rightarrow \infty . \end{aligned} \right. \end{aligned}$$
(1.10)

In turn, this is reduced to solving the integral equation

$$\begin{aligned} \mathbf{v }(z) = \mathbf{v }_{\ell }(z;\lambda ) - \lambda \int _{{\mathcal {E}}} \left( \mathbf{v }(z') \cdot \nabla _{z'} \right) \mathbf{G }(z,z';\lambda ) \cdot \mathbf{v }(z') \text {d}x' \text {d}y' =: T_\lambda \mathbf{v }, \end{aligned}$$
(1.11)

where \(\mathbf{G }\) is the Green tensor with Dirichlet boundary conditions for the linearized problem

$$\begin{aligned} \left\{ \begin{aligned}&\Delta \mathbf{v } - \lambda \partial _1 \mathbf{v } - \nabla q = 0, \\&\nabla \cdot \mathbf{v } = 0 \end{aligned} \right. \end{aligned}$$
(1.12)

in \({\mathcal {E}}\) and \(\mathbf{v }_{\ell }(z;\lambda )\) is the linear solution to (1.12) with the same boundary conditions as \(\mathbf{v }\) (both \(\mathbf{G }\) and \(\mathbf{v }_{\ell }\) were constructed and studied in [6]). It is proved in [7] that, when \(\lambda \) is sufficiently small (so that a crucial smallness estimate on \(\mathbf{v }_{\ell }\) is valid, see [7, Lemma 2.1] ), \(T_\lambda \) is a contraction mapping for \(\mathbf{v }\) belonging to a small ball centered at 0 in the Banach space

$$\begin{aligned} X_\lambda = \biggl \{\mathbf{v }\in C({{\mathcal {E}}},{{\mathbb {R}}}^2)\,:\, \Vert \mathbf{v }\Vert _{X_\lambda }:=\max _{z\in {\mathcal {E}}; i=1,2} \frac{|\mathbf{v }_i(z)|}{h_i(\lambda z)}<\infty \biggr \}. \end{aligned}$$
(1.13)

Then, using standard perturbative arguments, the existence and (local) uniqueness for fixed points of \(T_\lambda \) under the conditions of Theorem 1 can be obtained.

Despite many other efforts (see, e.g., [8, 10, 21]), the existence problem in (\(1.1_\lambda \)) at high Reynolds numbers (for arbitrary \(\lambda > 0\)) remains open. In the famous lecture by professor V.I. Yudovich, which he gave at the University of Cambridge and published in [32], this task was included in the list of the most important open problems in mathematical fluid mechanics.

Since the work [7], another interesting problem has appeared: whether uniqueness in Theorem 1 is true globally, that is, without assumption (1.7)? Note, that for the same flow problem in three dimensions, such a smallness condition is not needed to prove uniqueness, see, e.g., [5, 21]. In two dimensions, the problem appears to be much more difficult, mainly because the Dirichlet energy alone is not sufficient to control the behavior of functions at infinity. As shown in [10] Section XII.2, the usual energy estimate argument would run into immediate difficulties when applied to prove uniqueness for (\(1.1_\lambda \)) in the unbounded domain \({\mathcal {E}}\). We also point out that, when the Reynolds number \(\lambda \) is large, uniqueness is, in general, not expected to hold for the Navier–Stokes equations.Footnote 4 When \(\lambda = 0\), uniqueness of the trivial solution \(\mathbf{w }=0\) is conjectured by Amick in [1, p.99] and still remains an open problem.

It is well known (see, e.g., [7]), that every Finn–Smith solution has a finite Dirichlet integral, that is, it is a D-solution. The purpose of our article is to give a positive answer to the uniqueness problem for small Reynolds numbers in the class of D-solutions. The main result is as follows:

Theorem 2

There exists a positive constant \(\lambda _1\) depending only on the geometry of \(\partial {\mathcal {E}}\) such that, for \(0<\lambda <\lambda _1\) and for arbitrary D-solution \(\mathbf{w }\) to (\(1.1_\lambda \)), the identity \(\mathbf{w }(z) \equiv \mathbf{w }_{FS}(z;\lambda )\) holds. Here \(\mathbf{w }_{FS}(z;\lambda )\) is the Finn-Smith solution given by Theorem 1.

We prove Theorem 2 by deriving the estimate (1.7) for arbitrary D-solution to (\(1.1_\lambda \)) when \(\lambda \) is sufficiently small, thus invoking the uniqueness statement in Theorem 1. More precisely, Theorem 2 is an immediate corollary of the following lemma:

Lemma 3

There exist positive constants \(\lambda _1\) and \(M_1\) depending only on the geometry of \(\partial {\mathcal {E}}\) such that, for \(0<\lambda <\lambda _1\) and for arbitrary D-solution \(\mathbf{w }\) to (\(1.1_\lambda \)), we have

$$\begin{aligned} |(\mathbf{w } - \lambda \mathbf{e }_1)_i(z)| {\,\leqq \,}M_1 |\log \lambda |^{-\frac{1}{2}} \lambda h_i(\lambda z) {\,\leqq \,}{\varepsilon }_0 \lambda h_i(\lambda z), \ i = 1,2. \end{aligned}$$
(1.14)

Here \(\lambda _0\) and \({\varepsilon }_0\) are those given by Theorem 1.

Remark 4

In fact, we can prove Theorem 2 and Lemma 3 under more general boundary conditions for \(\mathbf{w }\) on \(\partial {\mathcal {E}}\). Namely, if \(\mathbf{w }|_{\partial {\mathcal {E}}} = \lambda \mathbf{a }\) and \(\mathbf{a }\) is a constant vector, then uniqueness as in Theorem 2 holds and our proof goes through with only minor modifications. See Remark 10 for further discussions.

Remark 5

The study of general D-solutions, that is, solutions to the Navier–Stokes system (\(1.1_\lambda \))\({}_{1,2}\) with a finite Dirichlet integral in \({{\mathcal {E}}}\) without any a priori conditions at infinity, was initialized by the Leray paper [23].Footnote 5 Many useful and elegant properties of D-solutions were discovered in the classical papers [1, 12]. Recently, based on these ideas, it was proved [18, 19] that every D-solution is uniformly bounded and, moreover, it has a constant uniform limit at infinity.Footnote 6 Of course, any D-solution \(\mathbf{w }\) to (\(1.1_\lambda \)) satisfies \(\mathbf{w } \rightarrow \lambda \mathbf{e }_1 \ne 0\) at infinity by assumption. In the remarkable paper of L.I. Sazonov [27] it was proved that any D-solution to (\(1.1_\lambda \)) satisfies rather strong decay estimates at infinity, namely, it is physically reasonable (or in the class of PR) in the sense of [29, page 350]. In [29], Smith showed that the behavior of any PR-solution is essentially controlled at infinity by that of the fundamental Oseen solution (see also [9] and [10, Ch. XII]). In particular, using the results of [27, 29], for any D-solution \(\mathbf{w }\) to (\(1.1_\lambda \)) we have

$$\begin{aligned} |(\mathbf{w } - \lambda \mathbf{e }_1)_i(z)| {\,\leqq \,}M(\lambda , \mathbf{w }) \lambda h_i(\lambda z), \ i = 1,2 \end{aligned}$$
(1.15)

for some constant \(M(\lambda , \mathbf{w })\). Thus, the essence of Lemma 3 is that the dependence of M on \(\mathbf{w }\) is removed and a smallness factor \(|\log \lambda |^{-\frac{1}{2}}\) is achieved when \(\lambda \) is small.

Note, that some uniqueness results concerning the 2d exterior problem were obtained recently, see, e.g., [15, 24, 33, 34], but in the quite different context and of quite different nature. Namely, in [24, 33, 34] it was considered the cases of zero limit at infinity under additional symmetry and smallness assumptions (including external force and energy inequality); and in [15] it was considered the case \({{\mathcal {E}}}={{\mathbb {R}}}^2\) under additional assumptions of type (1.7) and in the presence of external force.

More detailed survey of results concerning boundary value problems for the stationary NS-system in plane exterior domains can be found, for example, in [10, 13]. The subject is still a source of interest, as evidenced, e.g., by the papers [14,15,16].

Now let us describe the main ideas and approaches of our paper. The proof of Lemma 3 starts with an interesting and very useful result stating that any  D-solutions to (\(1.1_\lambda \)) have extra small Dirichlet energy when \(\lambda \) is small:

$$\begin{aligned} D_\lambda = \int _{\mathcal {E}} |\nabla \mathbf{w }|^2 {\,\leqq \,}C{\varepsilon }^2 \lambda ^2\quad \text{ with } \, {\varepsilon }=\frac{1}{\sqrt{|\log \lambda |}}. \end{aligned}$$
(1.16)

The presence of \({\varepsilon }\) is closely related to the Stokes paradox introduced earlier (see below Remark 10 for more detailed explanation of this fact). Note that similar logarithmic smallness was proved in [1, Theorem 22] for the Dirichlet energy of the Stokes solutions in bounded domains.

We now make a crucial observation: outside the critical circle \(\{r{\,\geqq \,}\frac{1}{\lambda }\}\), pressure is uniformly small, that is,

$$\begin{aligned} |p(z)|{\,\leqq \,}C{\varepsilon }\lambda ^2 \end{aligned}$$

(see Lemma 12). Also, we have to use a very elegant result of Amick [1] stating that Bernoulli pressure \(p+\frac{{\mathbf{w }}^2}{2}-\frac{\lambda ^2}{2}\) is increasing and decreasing along two regular vorticity level sets (curves) \(\{\omega =0\}\) which are going from \(\partial {{\mathcal {E}}}\) to infinity (see Lemma 13). The smallness of Dirichlet energy and pressure, along with the help of Amick’s technique of working with level sets, gives us many initial estimates concerning the behavior of \(\mathbf{w }\) outside (and near) the critical circle.

The estimates (1.14) are then considered separately, — inside and outside the critical circle \(\{r=\frac{1}{\lambda }\}\). The inequalities inside the critical circle can be obtained by applying standard technique based on estimates for solutions to the Stokes system inside disks, and it is a kind of routine (see the Appendix II).

Outside the critical circle, our circumstance and the techniques developed in papers [1, 12, 18, 20] allow to prove the uniform pointwise estimate

$$\begin{aligned} |{\mathbf{w }}(z)-(\lambda ,0)|{\,\leqq \,}C{\varepsilon }\lambda \end{aligned}$$
(1.17)

(see Section 5, Step 2 ). The last estimates open the possibility to apply the technique of the Sazonov paper [27] on integral operators associated with the fundamental solution to the Oseen system in order to prove the estimate

$$\begin{aligned} |{\mathbf{w }}(z)-(\lambda ,0)|{\,\leqq \,}C{\varepsilon }\lambda |\lambda z|^{-\frac{1}{4}-\delta }\ \qquad \forall |z|{\,\geqq \,}\frac{1}{\lambda }, \end{aligned}$$

which coincides with the standard inequality using in the definition of the PR-solutions. Finally the well-known results of Smith’s classical paper [29] allow to derive the required estimates (1.14) and to finish the proofs.

The rest of the paper is organized as follows. In Section 2, we introduce some frequently used notations and lemmas. A brief introduction to two-dimensional Oseen system is also included. In Section 3, a crucial smallness of Dirichlet energy is proved which has various interesting consequences. In Sections 4, 5 we prove pointwise bounds for \(\mathbf{w }\) inside and outside the critical circle respectively. The final proof of the main result is summarized at the end of Section 5. For a reader’s convenience, we moved the proof of the uniform estimates (1.17) (where the subtle real analysis arguments from [1] are used) into Appendix I.

2 Notations and Preliminaries

2.1 Notational Conventions

We work in the two dimensional setting and z will be a general point with Cartesian coordinates (x, y) in the plane. By a domain we mean an open connected set.

Let \(\Omega _{r_1, r_2} := \{z: r_1<|z|<r_2\}\) and \({\mathcal {E}}_r := \{z: |z| > r\}\). \(S_r\) will stand for the circle \(\{z: |z| = r\}\) and \(B_r\) for the open disk \(\{z: |z|< r\}\). We will often use \({\bar{f}}(r)\) to denote the average of a function f on the circle \(S_r\), that is,

$$\begin{aligned} {\bar{f}}(r) := \frac{1}{2\pi }\int _0^{2\pi } f(r,\theta ) \text {d}\theta . \end{aligned}$$

We use standard notations for Sobolev spaces \(W^{k,q}(\Omega )\), where \(k \in {\mathbb {N}}\), \(q\in [1,+\infty ]\). In our proof we do not distinguish the function spaces for scalar valued or vector valued functions, since it will be clear from the context which one we mean.

2.2 Properties of D-Solutions

We present some standard facts about the behavior of general D-functions (=functions with bounded Dirichlet integral). For the proof of the next Lemma, see, e.g., section 2 in [18].

Lemma 6

Let \(f \in W^{1,2}_{\mathrm {loc}(\Omega )}\) and assume that

$$\begin{aligned} D_f := \int _{\Omega _{r_1, r_2}} |\nabla f|^2 \text {d}x\text {d}y < \infty \end{aligned}$$

for some ring \(\Omega _{r_1, r_2} = \{ z \in {\mathbb {R}}^2 : 0< r_1< |z| < r_2 \} \subset \Omega \). Then we have

$$\begin{aligned} |{\bar{f}}(r_2) - {\bar{f}}(r_1)| {\,\leqq \,}\frac{1}{\sqrt{2\pi }}\left( D_f \ln \frac{r_2}{r_1}\right) ^\frac{1}{2}. \end{aligned}$$
(2.1)

Further more, if \(r_2 < \beta r_1\), then there exists a number \(r \in [r_1, r_2]\) such that

$$\begin{aligned} \sup _{|z| = r} |f(z) - {\bar{f}}(r)| {\,\leqq \,}C_\beta D_f^\frac{1}{2}, \end{aligned}$$
(2.2)

with constant \(C_\beta \) depending on \(\beta \) only.

The circles \(S_r\) in Lemma 6 will often be called good circles.

Next, we present an elegant lemma from [12, Theorem 4, page 399] that allows us to control the direction of the averaged velocity on circles for D-solutions to the Navier–Stokes equations.

Lemma 7

([12]) Let \(\mathbf{w }\) be a D-solution to the Navier–Stokes equations in some ring \(\Omega _{r_1, r_2} = \{z \in {\mathbb {R}}^2: 0< r_1<|z| < r_2\}\). Denote by \(\bar{\mathbf{w }}\) the average of \(\mathbf{w }\) over the circle \(S_r\) and let \(\varphi (r)\) be the argument of the complex number associated with the vector \(\bar{\mathbf{w }}(r) = ({\bar{w}}_1(r), {\bar{w}}_2(r))\), that is, \(\bar{\mathbf{w }}(r)=|{\bar{{\mathbf{w }}}}(r)|\,(\cos \varphi (r),\sin \varphi (r))\). Assume also that \(|\bar{\mathbf{w }}(r)| {\,\geqq \,}\sigma > 0\) for some constant \(\sigma \) and for all \(r \in (r_1, r_2)\). Then the estimate

$$\begin{aligned} \sup _{r_1< \rho _1 {\,\leqq \,}\rho _2 < r_2} |\varphi (\rho _2) - \varphi (\rho _1)| {\,\leqq \,}\frac{1}{4\pi \sigma ^2} \int _{\Omega _{r_1, r_2}} \left( \frac{1}{r}|\nabla \omega | + |\nabla \mathbf{w }|^2 \right) \end{aligned}$$
(2.3)

holds. Here, \(\omega = \partial _2 w_1 - \partial _1 w_2\) is the vorticity.

2.3 The Stokes Estimates

We recall the following classical local regularity estimate for the linear Stokes system (for the proof, see, for instance, [10, Theorem IV.4.1 and Remark IV.4.1]):

Lemma 8

Let \(\mathbf{w }_S\) be a local solution in \(B_1\) to the Stokes system

$$\begin{aligned} \left\{ \begin{aligned}&\Delta \mathbf{w }_S - \nabla p_S = \mathbf{f }_S, \\&\nabla \cdot \mathbf{w }_S = 0. \\ \end{aligned} \right. \end{aligned}$$
(2.4)

Then there holds the following regularity estimates for \(k=0,1,2,\cdots \) and \(1<s<\infty \):

$$\begin{aligned} \Vert \nabla ^{k+2} \mathbf{w }_S\Vert _{L^s(B_\frac{1}{2})} + \Vert \nabla ^{k+1} p_S\Vert _{L^s(B_\frac{1}{2})} {\,\leqq \,}C(k,s) \,\bigl (\Vert \mathbf{w }_S\Vert _{W^{1,s}(B_1)} + \Vert {\mathbf{f }}_S\Vert _{W^{k,s}(B_1)}\bigr ). \end{aligned}$$
(2.5)

Moreover, the domains \(B_\frac{1}{2}\) can be replaced by \(B^+_\frac{1}{2} = B_\frac{1}{2} \cap {\mathbb {R}}^2_+\) if we assume that \(\mathbf{w }_S = 0\) on \(\partial {{\mathbb {R}}^2_+} \cap \partial B^+_\frac{1}{2}\). Here \({\mathbb {R}}^2_+\) is the upper half plane \(\{(x,y): y>0\}\).

It follows from this lemma and standard bootstrapping arguments that D-solutions to (\(1.1_\lambda \)) are locally smooth.

2.4 The Oseen System

For convenience of our presentation in Section 5, here we summarize some known results on the Oseen system (1.12) in two dimensions. The fundamental solution of the Ossen system \((\mathbf{E },\mathbf{e })\), introduced in [25], consists of a symmetric tensor of rank two \(E_{ij}\) and a vector \(e_j\), such that

$$\begin{aligned} \Delta E_{ij} - \partial _1 E_{ij} - \partial _i e_j&= \delta _{ij} \delta _0, \nonumber \\ \sum _{i=1,2} \partial _i E_{ij}&= 0, \end{aligned}$$
(2.6)

where \(i, j = 1, 2\) and \(\delta _0\) is the delta function supported at the origin. Explicitly, \((\mathbf{E }, \mathbf{e })\) are given by

$$\begin{aligned} \mathbf{E } = \left[ \begin{matrix} \partial _1(H+L) - L \ &{}\partial _2 (H + L)\\ \partial _2(H + L) \ &{}-\partial _1(H+L) \end{matrix}\right] , \quad \mathbf{e } = -\nabla H \end{aligned}$$
(2.7)

where \(\Delta H = \delta _0\) and \(- \Delta L + \partial _1 L = \delta _0\). More explicitly, H and L are given by

$$\begin{aligned} H = \frac{1}{2\pi } \ln r, \quad L = \frac{1}{2\pi } e^{r\cos \theta /2} K_0(r/2) \end{aligned}$$
(2.8)

where \(K_0\) denotes the modified Bessel function of the second kind. Asymptotically, it holds that

$$\begin{aligned} K_0(\rho ) = \sqrt{\frac{\pi }{2}} \left( \frac{1}{\rho ^{1/2}} + O\left( \frac{1}{\rho ^{3/2}}\right) \right) e^{-\rho } \end{aligned}$$
(2.9)

as \(\rho \rightarrow \infty \). As a consequence, \(E_{11}\) exhibits a parabolic wake region \(\{(x,y): x {\,\geqq \,}0, |y| {\,\leqq \,}\sqrt{x} \}\) in which the decay at infinity is slower than outside. It is also known that near the singularity of E at \(z=0\), it holds that

$$\begin{aligned} E_{ij}(z) = - \frac{1}{4\pi } \left( \delta _{ij} \ln \frac{1}{r} + \frac{z_iz_j}{r^2} \right) + o(1) \end{aligned}$$
(2.10)

as \(r = |z|\rightarrow 0\). Here \(z_1 = x\), \(z_2 = y\). The Fourier transform of \(\mathbf{E }\) is given by

$$\begin{aligned} {\hat{E}}_{ij}(\xi ) = \frac{\xi _i\xi _j - |\xi |^2 \delta _{ij}}{|\xi |^2 (|\xi |^2 + i\xi _1)}. \end{aligned}$$
(2.11)

More detailed asymptotic behavior and summability of (E, e) are summarized in [10, Section VII.3]. Next, we write \(\mathbf{T }(\mathbf{u }, p)\) for the stess tensor

$$\begin{aligned} \mathbf{T }(\mathbf{u },p) = \nabla \mathbf{u } + (\nabla \mathbf{u })^\intercal - p \mathbf{I }. \end{aligned}$$
(2.12)

It is straightforward to check the following Green identity for the Oseen operator using integration by parts:

$$\begin{aligned}&\int _\Omega (\nabla \cdot \mathbf{T }(\mathbf{u },p) + \partial _1 \mathbf{u }) \cdot \mathbf{v } - \int _\Omega (\nabla \cdot \mathbf{T }(\mathbf{v },q) - \partial _1 \mathbf{v }) \cdot \mathbf{u } \nonumber \\&\quad \quad = \int _{\partial \Omega } \mathbf{v } \cdot \mathbf{T }(\mathbf{u },p) \cdot \mathbf{n } - \mathbf{u } \cdot \mathbf{T }(\mathbf{v },q) \cdot \mathbf{n } + (\mathbf{u } \cdot \mathbf{v }) (\mathbf{n } \cdot \mathbf{e }_1), \end{aligned}$$
(2.13)

for any pairs \((\mathbf{u },p), (\mathbf{v },q)\) such that \(\mathbf{u },\mathbf{v }\) are smooth solenoidal vector fields and p, q are smooth scalar functions in \({\bar{\Omega }}\). Here \(\mathbf{n } = (n_1, n_2)\) is the normal vector of \(\partial {\Omega }\) pointing outward with respect to \(\Omega \). Suppose that \((\mathbf{v },q)\) is a solution to the forced Oseen system

$$\begin{aligned} \Delta \mathbf{v } - \partial _1 \mathbf{v } - \nabla q&= \mathbf{f },\\ \nabla \cdot \mathbf{v }&= 0, \end{aligned}$$

in a bounded domain \(\Omega \) with sufficiently smooth boundary. Using (2.13) for \((\mathbf{E }(z - \cdot ), -\mathbf{e }(z - \cdot ))\) and \((\mathbf{v }, q)\) we have

$$\begin{aligned} \mathbf{v }(z)&= \int _\Omega \mathbf{f }(z') \mathbf{E }(z - z') \nonumber \\&-\int _{\partial \Omega } \mathbf{v }(z') \cdot \mathbf{T }_z(\mathbf{E }, \mathbf{e })(z-z') \cdot \mathbf{n } \nonumber \\&- \int _{\partial \Omega } \mathbf{E }(z-z') \cdot \mathbf{T }(\mathbf{v },q)(z') \cdot \mathbf{n } + \int _{\partial \Omega } (\mathbf{v }(z') \cdot \mathbf{E }(z-z')) (\mathbf{n } \cdot \mathbf{e }_1) \end{aligned}$$
(2.14)

for any \(z \in {\bar{\Omega }}\). Here the integrals are taken over the variable \(z'\) and \(\mathbf{T }_z\) means that we take derivative in z when defining \(\mathbf{T }\). For any D-solution \(\mathbf{w }\) to the Navier–Stokes equations in the exterior domain \({\mathcal {E}}\) tending to the limiting velocity \(\mathbf{e }_1\) at infinity, the following representation formula holds in \({\mathcal {E}}\) for \(\mathbf{v } = \mathbf{w } - \mathbf{e }_1\):

$$\begin{aligned} \mathbf{v }(z)&= - \int _{{\mathcal {E}}} (\mathbf{v } \cdot \nabla _{z'}) \mathbf{E }(z - z') \cdot \mathbf{v } \nonumber \\&-\int _{\partial {\mathcal {E}}} \mathbf{v }(z') \cdot \mathbf{T }_z(\mathbf{E }, \mathbf{e })(z-z') \cdot \mathbf{n } \nonumber \\&- \int _{\partial {\mathcal {E}}} \mathbf{E }(z-z') \cdot \mathbf{T }(\mathbf{v },q)(z') \cdot \mathbf{n } + \int _{\partial {\mathcal {E}}} (\mathbf{v }(z') \cdot \mathbf{E }(z-z')) (\mathbf{n } \cdot \mathbf{w }(z')). \end{aligned}$$
(2.15)

In [29], pointwise asymptotic behavior of \(\mathbf{v }\) is obtained through this representation formula for solutions of class PR. For more details, we refer to [29, Theorem 5] .

3 Smallness of \(\pmb {D_\lambda }\) and the Corresponding Estimates for Pressure and Bernoulli Pressure

From here and for the subsequent three sections, we shall always let \(\mathbf{w }\) be an arbitrary D-solution to (\(1.1_\lambda \)) with \(\lambda >0\).

Unless otherwise specified, we use C, M to denote absolute positive constants, or positive constants that depend only on \(\partial {\mathcal {E}}\). It is important that they do not depend on \(\lambda \). The specific value of such constants may change from line to line.

Denote the total Dirichlet energy of \(\mathbf{w }\) by \(D_\lambda \). Using the celebrated reductio ad absurdum argument of Leray [23] or the Leray-Hopf extension method [17], it is possible to prove that, for any \(0<\lambda <\Lambda \), the apriori bound \(D_\lambda < M_3(\Lambda , \partial {\mathcal {E}})\) holds for some constant \(M_3\). It turns out, however, that when \(\lambda \) is small, we can use a special solenoidal extension of the boundary value in (\(1.1_\lambda \)) to prove an extra smallness of the Dirichlet energy. Note, that such an extension was also used earlier in [1, Theorem 22] to study the Dirichlet energy of the Stokes solutions in bounded domains.

The main result of the section is the following lemma:

Lemma 9

There exist constants \(0<\lambda _2<\frac{1}{2}, M_2>0\) depending only on the geometry of \(\partial {\mathcal {E}}\) such that, for \(0< \lambda < \lambda _2\), we have

$$\begin{aligned} D_\lambda = \int _{\mathcal {E}} |\nabla \mathbf{w }|^2 {\,\leqq \,}\frac{M_2\lambda ^2}{|\log \lambda |}. \end{aligned}$$
(3.1)

Proof

Let \(\tau \in C^\infty ({\mathbb {R}})\) with \(\tau (r) = 0\) for \(r{\,\leqq \,}\frac{1}{2}\) and \(\tau (r) = 1\) for \(r{\,\geqq \,}1\). Define \(\mu (r) = \tau (\log r/ \log R)\), so that \(\mu (r) = 0\) when \(r{\,\leqq \,}\sqrt{R}\). The parameter \(R{\,\geqq \,}2\) has to be chosen later. Define a solenoidal vector field \(\mathbf{A } = (A_1, A_2)\) by

$$\begin{aligned} A_1 = \partial _y (\lambda y \mu (|z|)), \quad A_2 = -\partial _x (\lambda y \mu (|z|)). \end{aligned}$$

Such \(\mathbf{A }\) clearly satisfies the same boundary condition as in (\(1.1_\lambda \)). Set \(\tilde{\mathbf{w }}=\mathbf{w }-\mathbf{A }\), then \(\tilde{\mathbf{w }} = 0\) on \(\partial {\mathcal {E}}\) and \(\tilde{\mathbf{w }} \rightarrow 0\) at \(\infty \). By (\(1.1_\lambda \)), \(\tilde{\mathbf{w }} \) satisfies the equation

$$\begin{aligned} -\Delta \tilde{\mathbf{w }} - \Delta \mathbf{A } + (\tilde{\mathbf{w }} +\mathbf{A })\cdot \nabla \tilde{\mathbf{w }} + \tilde{\mathbf{w }} \cdot \nabla \mathbf{A } + \mathbf{A } \cdot \nabla \mathbf{A } + \nabla p =0. \end{aligned}$$

Multiplying this equation by \(\tilde{\mathbf{w }} \) and integrating in \({\mathcal {E}}\) gives

$$\begin{aligned} \int _{\mathcal {E}} |\nabla \tilde{\mathbf{w }} |^2 + \int _{\mathcal {E}} \nabla \mathbf{A } \cdot \nabla \tilde{\mathbf{w }} + \int _{\mathcal {E}} (\tilde{\mathbf{w }} \cdot \nabla ) \mathbf{A } \cdot \tilde{\mathbf{w }} + \int _{\mathcal {E}} (\mathbf{A } \cdot \nabla ) \mathbf{A } \cdot \tilde{\mathbf{w }} =0. \end{aligned}$$
(3.2)

(Note that we actually first carry out this energy estimate in \({\mathcal {E}} \cap B_\rho \) and then let \(\rho \rightarrow \infty \). Due to the asymptotic behaviour of \(\mathbf{w }\) at infinity proved in [29, Theorem 5]  (see estimates (1.15) from our Remark 5), the boundary integrals on \(S_\rho \) converge to 0, thus leading to (3.2).)

The constructed extension \(\mathbf{A }\) satisfies the pointwise bounds

$$\begin{aligned} |\mathbf{A }|{\,\leqq \,}C \lambda , \quad |\nabla \mathbf{A }|^2 {\,\leqq \,}\frac{C\lambda ^2}{|z|^2 (\log R)^2}. \end{aligned}$$

As a consequence, the Dirichlet energy of \(\mathbf{A }\) is bounded by

$$\begin{aligned} \int _{\mathcal {E}} |\nabla \mathbf{A }|^2 {\,\leqq \,}\frac{C\lambda ^2}{\log R}. \end{aligned}$$
(3.3)

The second term in (3.2) can be treated with Hölder’s inequality and (3.3), while the fourth term in (3.2) is estimated by

$$\begin{aligned} -\int (\mathbf{A } \cdot \nabla ) \mathbf{A } \cdot \tilde{\mathbf{w }}&= \int {(\mathbf{A } \cdot \nabla ) \tilde{\mathbf{w }}} \cdot (\mathbf{A }-\mathbf{w }_\infty ) \nonumber \\&{\,\leqq \,}\frac{1}{4} \int |\nabla \tilde{\mathbf{w }}|^2 + \int _{r{\,\leqq \,}R} |A|^2 |A-\mathbf{w }_\infty |^2 \nonumber \\&{\,\leqq \,}\frac{1}{4} \int |\nabla \tilde{\mathbf{w }}|^2 + C \lambda ^4 R^2 . \end{aligned}$$
(3.4)

The use of (3.3) gives a bound for the third term in (3.2),

$$\begin{aligned} -\int (\tilde{\mathbf{w }} \cdot \nabla \mathbf{A }) \cdot \tilde{\mathbf{w }}&{\,\leqq \,}\left( \int _{r {\,\leqq \,}R} |\tilde{\mathbf{w }}|^4\right) ^\frac{1}{2} \left( \int |\nabla \mathbf{A }|^2 \right) ^\frac{1}{2} \nonumber \\&{\,\leqq \,}\frac{C \lambda }{\sqrt{\log R}} \left( \int _{r {\,\leqq \,}R} |\tilde{\mathbf{w }}|^4\right) ^\frac{1}{2} \nonumber \\&{\,\leqq \,}\frac{C \lambda }{\sqrt{\log R}} \cdot R^2 \left( \int |\nabla \tilde{\mathbf{w }}|^2\right) . \end{aligned}$$
(3.5)

In the last step we used the following Sobolev type inequality in \(\Omega _R = B_R \cap {\mathcal {E}}, R{\,\geqq \,}2\) for functions f defined in \(\Omega _R\) with the property that f vanishes on the boundary \(\partial {\mathcal {E}}\):

$$\begin{aligned} \Vert f\Vert _{L^4(\Omega _R)} {\,\leqq \,}CR \Vert \nabla f\Vert _{L^2(\Omega _R)}. \end{aligned}$$

This can be easily checked using Sobolev imbeddings for unit disk, scaling, and the inequality (2.1).

Now we let \(R = \lambda ^{-1/4}\). Choose \(\lambda _2<\frac{1}{2}\) sufficiently small such that for any \(0<\lambda <\lambda _2\) we have

$$\begin{aligned} \frac{C R^2 \lambda }{\sqrt{\log R}} = \frac{2C \lambda ^{1/2}}{\sqrt{|\log \lambda |}}< \frac{1}{4}, \qquad \ \ \ \ \ \lambda ^4 R^2=\lambda ^{3\frac{1}{2}}<\frac{\lambda ^2}{\sqrt{\log \lambda }}. \end{aligned}$$

Then, combining the estimates (3.2), (3.3), (3.4), and (3.5), we obtain for \(0<\lambda <\lambda _2\) that

$$\begin{aligned} \int |\nabla \tilde{\mathbf{w }}|^2 {\,\leqq \,}\frac{C \lambda ^2}{|\log \lambda |}. \end{aligned}$$
(3.6)

Together with (3.3), inequality (3.6) gives the conclusion. \(\square \)

Remark 10

The small factor \(|\log \lambda |^{-1}\) will be essential for our remaining arguments. Its presence is, in some sense, a reflection of the Stokes paradox. For example, it guarantees that for any sequence of solutions \(\mathbf{w }^{(k)}\) to (\(1.1_\lambda \)) with \(\lambda _k \rightarrow 0\), the modified sequence \(\lambda _k^{-1} \mathbf{w }^{(k)}\), which is solution to the system

$$\begin{aligned} \left\{ \begin{aligned}&\Delta \mathbf{w } - \lambda _k(\mathbf{w } \cdot \nabla ) \mathbf{w } - \nabla p = 0, \\&\nabla \cdot \mathbf{w } = 0, \\&\mathbf{w }|_{\partial {\mathcal {E}}} = 0, \\&\mathbf{w }(z) \rightarrow \mathbf{e }_1 \ \ \text {as}\ \ |z| \rightarrow \infty , \end{aligned} \right. \end{aligned}$$
(3.7)

converges (on every bounded set) to the zero function which is the only D-solution to the Stokes system (1.1)\(_{1,2,3}\). If, instead of \(\mathbf{w }|_{\partial {\mathcal {E}}} = 0\) in (\(1.1_\lambda \)), one prescribes a more general boundary condition \(\mathbf{w }|_{\partial {\mathcal {E}}} = \lambda \mathbf{a }\) for a general fixed non-constant function \(\mathbf{a }\), then due to [7, Theorem 4.2], one can still construct solutions to the Navier–Stokes equations when \(\lambda \) is sufficiently small. However, for such solutions only a weaker bound \(\int _{\mathcal {E}} |\nabla \mathbf{w }|^2 < M_4 \lambda ^2\) can be obtained, see [7, Lemma 5.2]. Thus the uniqueness (or non-uniqueness) of solutions under such inhomogeneous boundary conditions among all D-solutions is a more subtle problem, and lies still beyond the reach of our methods. Nevertheless, when \(\mathbf{a }\) is a fixed constant vector on \(\partial {\mathcal {E}}\), the proof of Lemma 9 works by a slight modification in the definition of \(\mathbf{A }\). The rest of our proof (including the uniqueness result) also works in this situation.

Next, using (\(1.1_\lambda \)), (3.1), Lemma 8, and standard bootstrapping arguments, we obtain explicit bounds for \(\mathbf{w }\) near \(\partial {\mathcal {E}}\), namely, in \(\Omega _3 = {\mathcal {E}} \cap B_3\). For simplicity, we work with infinitely smooth \(\partial {{\mathcal {E}}}\) here. If \(\partial {{\mathcal {E}}}\) has only finite regularity, then the following estimates near the boundary are valid up to a finite k (nevertheless still sufficient for the rest of the paper):

Lemma 11

Let \(\mathbf{w }\) be an arbitrary D-solution to (\(1.1_\lambda \)) for some \(0< \lambda < \lambda _2\). Then \(\Vert \mathbf{w }\Vert _{C^{k}(\Omega _{3})} {\,\leqq \,}C(k,\partial {\mathcal {E}}) D_\lambda ^{\frac{1}{2}}\) for any integer \(k {\,\geqq \,}0\). Moreover, up to the subtraction of a suitable constant, the associated pressure \(p \rightarrow 0\) at infinity and \(\Vert p\Vert _{C^{k}(\Omega _{3})} {\,\leqq \,}C(k, \partial {\mathcal {E}}) D_\lambda ^{\frac{1}{2}}\) for any integer \(k {\,\geqq \,}0\).Footnote 7

Proof

The regularity estimate of \(\mathbf{w }\) is standard. We have used that \(D_\lambda \) is small, so that \(D_\lambda ^{\frac{m}{2}}, m {\,\geqq \,}2\) coming from the nonlinear term are dominated by \(D_\lambda ^{\frac{1}{2}}\). Let us explain the last statement. It is proved in [12] that the pressure p has a uniform limit at infinity which, after the subtraction of a suitable constant, may be taken as 0. Denote \({\bar{p}}(r) = \frac{1}{2\pi } \int _0^{2\pi } p(r, \theta ) \text {d}\theta \). It is shown in [12, Lemma 4.1] that

$$\begin{aligned} 2\pi |{\bar{p}}(r_2) - {\bar{p}}(r_1)| {\,\leqq \,}\int _{r>r_1} |\nabla \mathbf{w }|^2 \text {d}x\text {d}y {\,\leqq \,}D_\lambda \end{aligned}$$
(3.8)

for any \(1 {\,\leqq \,}r_1 {\,\leqq \,}r_2 < \infty \). Sending \(r_2 \rightarrow \infty \), (3.8) gives, for any \(r_1 {\,\geqq \,}1\),

$$\begin{aligned} |{\bar{p}}_{r_1}| {\,\leqq \,}\frac{D_\lambda }{2\pi }. \end{aligned}$$
(3.9)

Note that \(\Vert \nabla p\Vert _{C^k(\Omega _{3})} {\,\leqq \,}C(k, \partial {\mathcal {E}}) D_\lambda ^{\frac{1}{2}}\) follows from (2.5) and bootstrapping arguments. Using (3.9), we get

$$\begin{aligned} \Vert p\Vert _{C^{k+1}(\Omega _{3})}&{\,\leqq \,}C(k, \partial {\mathcal {E}}) (D_\lambda ^{\frac{1}{2}} + D_\lambda ) \nonumber \\&{\,\leqq \,}C(k, \partial {\mathcal {E}}) D_\lambda ^{\frac{1}{2}}. \end{aligned}$$
(3.10)

\(\square \)

The next lemma plays the crucial role in many estimates near and outside the critical circle \(|z|=\frac{1}{\lambda }\).

Lemma 12

Let \(\mathbf{w }\) be an arbitrary D-solution to (\(1.1_\lambda \)) for some \(0< \lambda < \lambda _2\). Then the pressure p can be decomposed as \(p = p_1 + p_2\) such that

$$\begin{aligned}&\lim _{z \rightarrow \infty } p_1 = \lim _{z \rightarrow \infty } p_2 = 0, \end{aligned}$$
(3.11)
$$\begin{aligned}&\quad \quad \Vert p_1\Vert _{C^0} {\,\leqq \,}C D_\lambda , \end{aligned}$$
(3.12)
$$\begin{aligned}&|p_2(z)|{\,\leqq \,}\frac{CD_\lambda ^{\frac{1}{2}}}{|z|}, \quad \forall |z| {\,\geqq \,}2. \end{aligned}$$
(3.13)

Proof

By equations (\(1.1_\lambda \))\(_1\) and (\(1.1_\lambda \))\(_2\), the pressure solves the Poisson equation in \({\mathcal {E}}\):

$$\begin{aligned} \Delta p = -\nabla \mathbf{w } \cdot (\nabla \mathbf{w })^\intercal . \end{aligned}$$
(3.14)

Let \(p_1\) be the potential solution to (3.14), that is,

$$\begin{aligned} p_1(z) = - \frac{1}{2 \pi } \int _{{\mathcal {E}}} \log |z-\zeta | (\nabla \mathbf{w } \cdot (\nabla \mathbf{w })^\intercal )(\zeta ) \, \text {d}\zeta _1\, \text {d}\zeta _2. \end{aligned}$$

By the classical div-curl lemma (see, e.g., [3]),  \(\nabla \mathbf{w } \cdot (\nabla \mathbf{w })^\intercal \) belongs to the Hardy space \({\mathcal {H}}^1({\mathbb {R}}^2)\). Hence, by the Calderón-Zygmund theorem for Hardy spaces [30], \(\nabla ^2 p_1 \in L^1({\mathbb {R}}^2)\), and \(\nabla p \in L^2({\mathbb {R}}^2)\). Moreover, \(p_1 \in C^0({\mathbb {R}}^2)\) and converges to 0 at infinity. In particular,

$$\begin{aligned} \sup _{{\mathbb {R}}^2} |p_1| {\,\leqq \,}C \Vert \nabla \mathbf{w } \cdot (\nabla \mathbf{w })^\intercal \Vert _{{\mathcal {H}}^1} {\,\leqq \,}C D_\lambda . \end{aligned}$$
(3.15)

Let \(p_2 = p - p_1\), a function defined in \({\mathcal {E}}\). Clearly, \(p_2\) is a harmonic function and satisfies

$$\begin{aligned} \lim _{r\rightarrow \infty } p_2 = 0. \end{aligned}$$
(3.16)

By Lemma 11 and (3.15), we have

$$\begin{aligned} \quad \sup _{S_1}|p_2|&{\,\leqq \,}\sup _{S_1} (|p| + |p_1|) \nonumber \\&{\,\leqq \,}C (D_\lambda ^{\frac{1}{2}} + D_\lambda ) \nonumber \\&{\,\leqq \,}C D_\lambda ^{\frac{1}{2}}. \end{aligned}$$
(3.17)

Since \(z \mapsto \frac{1}{z}\) is a conformal mapping in the extended complex plane, \(p_2(\frac{1}{z})\) is a harmonic function in \(z \in B_1\) with \(p_2(\frac{1}{0}) = 0\). Then by the classical Schwarz lemma and by estimate (3.17), we have \(|p_2(\frac{1}{z})| {\,\leqq \,}C D_\lambda ^{\frac{1}{2}} |z|\) for \(|z| {\,\leqq \,}\frac{1}{2}\). This proves (3.13). \(\square \)

Let \(\Phi = \frac{|\mathbf{w }|^2}{2} + p\) be the Bernoulli function. To proceed, we need an elegent observation made by Amick [1] for a general D-solution \(\mathbf{w }\) to (\(1.1_\lambda \))\(_{1,2,3}\) in \({\mathcal {E}}\). This result concerns topological behaviour of certain level sets of the vorticity \(\omega = \partial _2 w_1 - \partial _1 w_2\). Namely, Amick proved that there exist finitely many distinct unbounded connected components of the set \(\{z\in {{\mathcal {E}}}: \omega (z) \ne 0\}\). These components are denoted by \(V_+\) and \(V_-\) in his paper [1]. Each \(V_+\) and \(V_-\) is a simply-connected domain not separated from \(\partial {\mathcal {E}}\), essentially due to the maximum principle for \(\omega \). Then one may take the two unbounded components of \(\partial V_+\) for some \(V_+\) as \({\mathcal {C}}_i, \ i=1,2\), so that the statements in the following lemma are satisfied:

Lemma 13

(See [1] Theorem 11) For any D-solution \(\mathbf{w }\) to (\(1.1_\lambda \)), there exist two unbounded continuous curves \({\mathcal {C}}_i\), parametrized by arc length as \({\mathcal {C}}_i = \{(x_i(s), y_i(s): s \in (0, \infty ))\}, \, i=1, 2\). The functions \(x_i(\cdot )\) and \(y_i(\cdot )\) are real-analytic on \((0, \infty )\) except possibly at isolated points, and they satisfy \((x_i(0), y_i(0))\) \(\in \{|z| = 1\}\) and \(|(x_i(s), y_i(s))| \rightarrow \infty \) as \(s\rightarrow \infty \). The vorticity \(\omega \) vanishes on these two curves. Moreover, the maps \(s \rightarrow \Phi (x_i(s), y_i(s))\) are monotone decreasing and increasing in \(s \in (0, \infty )\) respectively for \(i=1,2\).

Two immediate consequences of Lemma 13 and the known fact that \(\Phi \rightarrow \frac{\lambda ^2}{2}\) at infinity are as follows: for any \(r{\,\geqq \,}1\) we have

$$\begin{aligned} \max _{|z|=r} \Phi (z) {\,\geqq \,}\frac{\lambda ^2}{2}, \end{aligned}$$
(3.18)

and

$$\begin{aligned} \min _{|z|=r} \Phi (z) {\,\leqq \,}\frac{\lambda ^2}{2}. \end{aligned}$$
(3.19)

Hence, using Lemma 12, for any \(r = |z| {\,\geqq \,}\lambda ^{-1}\), it holds that

$$\begin{aligned} \max _{|z| = r} |\mathbf{w }|^2 {\,\geqq \,}\lambda ^2 - C\lambda D_\lambda ^{\frac{1}{2}}, \end{aligned}$$
(3.20)

and

$$\begin{aligned} \min _{|z| = r} |\mathbf{w }|^2 {\,\leqq \,}\lambda ^2 + C\lambda D_\lambda ^{\frac{1}{2}} \end{aligned}$$
(3.21)

for arbitrary D-solution \(\mathbf{w }\) to (\(1.1_\lambda \)) with \(0< \lambda < \lambda _2\).

Due to Lemma 6, there exists a sequence of “good” radii \(R_n \in [2^n \lambda ^{-1}, 2^{n+1}\lambda ^{-1})\) for \(\ n=-4, -3, \cdots ,1,2,\cdots \), such that

$$\begin{aligned} |\mathbf{w }(R_n, \theta ) - \bar{\mathbf{w }}(R_n)| {\,\leqq \,}C D_\lambda ^{\frac{1}{2}} \end{aligned}$$
(3.22)

for all \(0 {\,\leqq \,}\theta < 2\pi \). From (3.20), (3.21) and by the triangle inequality we obtain

$$\begin{aligned} \big | |\bar{\mathbf{w }}(R_n)| - \lambda \big | {\,\leqq \,}CD_\lambda ^{\frac{1}{2}}. \end{aligned}$$
(3.23)

Hence, by (2.1) of Lemma 6, we have

$$\begin{aligned} \Big | |\bar{\mathbf{w }}(r)| - \lambda \Big | {\,\leqq \,}CD_\lambda ^\frac{1}{2} \end{aligned}$$
(3.24)

for any \(r {\,\geqq \,}\frac{1}{16\lambda }\). When \(\lambda \) is sufficiently small, this implies, in particular, that

$$\begin{aligned} |\bar{\mathbf{w }}(r)| {\,\geqq \,}\frac{\lambda }{2}. \end{aligned}$$
(3.25)

In order to use Lemma 7 to control the direction of the vector \(\bar{\mathbf{w }}\), below we need to establish some suitable estimates for \(\mathbf{w }\) in the region \(r > rsim \lambda ^{-1}\).

Lemma 14

Let \(\mathbf{w }\) be an arbitrary D-solution to (\(1.1_\lambda \)) with \(0< \lambda < \min \{\lambda _2, \frac{1}{16}\}\). Then the estimate \(|\mathbf{w }| {\,\leqq \,}C\lambda \) holds for all \(r {\,\geqq \,}(8\lambda )^{-1}\).

Proof

By Lemma 12, we have \(|p| {\,\leqq \,}C\lambda ^2 |\log \lambda |^{-\frac{1}{2}}\) for \(r {\,\geqq \,}(16\lambda )^{-1}\). According to the estimates (3.1) and (3.22), (3.23), we obtain

$$\begin{aligned} \Phi {\,\leqq \,}C\lambda ^2 \end{aligned}$$
(3.26)

on the good circle \(S_{R_{-4}}\). Recall, that \(\Phi \) satisfies the classical identity

$$\begin{aligned} \Delta \Phi = \omega ^2+\mathbf{w } \cdot \nabla \Phi . \end{aligned}$$

Therefore, from the maximum principle for \(\Phi \) and from the convergence \(\Phi \rightarrow \frac{\lambda ^2}{2}\) at infinity, we obtain \(\Phi {\,\leqq \,}C\lambda ^2\) in the region \(r {\,\geqq \,}R_{-4}\). Combined with the mentioned estimate of p, this clearly implies \(|\mathbf{w }| < C\lambda \) in the region \(r {\,\geqq \,}(8\lambda )^{-1}\). \(\square \)

Lemma 15

Let \(\mathbf{w }\) be as in Lemma 14, then \(\int _{{\mathcal {E}} \backslash B_{(4\lambda )^{-1}}} r|\nabla \omega |^2 \,\text {d}x\text {d}y {\,\leqq \,}C\lambda D_\lambda \).

Proof

Let \(\rho _*\in ((8\lambda )^{-1}, (4\lambda )^{-1})\) to be chosen later. The classical vorticity equation \(\Delta \omega =\mathbf{w } \cdot \nabla \omega \) implies the identity

$$\begin{aligned} \hbox {div}\,(r\omega \nabla \omega )=r|\nabla \omega |^2+\omega \partial _r\omega -(\mathbf{w }\cdot \mathbf{e }_r)\frac{\omega ^2}{2}+\hbox {div}\,\bigl (r\mathbf{w }\frac{\omega ^2}{2}\bigr ). \end{aligned}$$
(3.27)

Then we have the energy estimate

$$\begin{aligned}&\int _{r{\,\geqq \,}\rho _*} r|\nabla \omega |^2 \text {d}x\text {d}y + \int _{r {\,\geqq \,}\rho _*} \omega \partial _r \omega \,\text {d}x\text {d}y - \int _{r{\,\geqq \,}\rho _*} (\mathbf{w }\cdot \mathbf{e }_r )\frac{\omega ^2}{2} \,\text {d}x\text {d}y \nonumber \\&+\, \rho _* \int _{S_{\rho _*}} \partial _r \frac{\omega ^2}{2} ds - \rho _* \int _{S_{\rho _*}} (\mathbf{w }\cdot \mathbf{e }_r ) \frac{\omega ^2}{2} ds = 0. \end{aligned}$$
(3.28)

(Strictly speaking, to obtain the last formula, we have to integrate (3.27) on bounded domains \(B_\rho \setminus B_{\rho _*}\) first and then let \(\rho \rightarrow +\infty \), that is, the outer boundary goes to infinity. Using boundedness of Dirichlet energy of \(\mathbf{w }\), it can be easily checked that the boundary terms on large circles \(|z|=\rho \) disappear, at least for a sequence of radii going to infinity.)

The second term in (3.28) can be treated using Hölder’s inequality as follows:

$$\begin{aligned} \left| \int _{r{\,\geqq \,}\rho _*} \omega \partial _r \omega \,\text {d}x\text {d}y \right|&{\,\leqq \,}\frac{1}{2\rho _*} \int _{r{\,\geqq \,}\rho _*} \omega ^2 \,\text {d}x\text {d}y + \frac{\rho _*}{2} \int _{r {\,\geqq \,}\rho _*} |\partial _r \omega |^2 \,\text {d}x\text {d}y \nonumber \\&{\,\leqq \,}\frac{D_\lambda }{\rho _*} + \frac{1}{2} \int _{r {\,\geqq \,}\rho _*} r|\nabla \omega |^2. \end{aligned}$$

The third term on the left of (3.28) is controlled by \(C \lambda D_\lambda \) since by Lemma 14\(|\mathbf{w }| {\,\leqq \,}C\lambda \) holds true in \({\mathcal {E}} \backslash B_{(8\lambda )^{-1}}\). Hence, we just need to treat the boundary terms. Using

$$\begin{aligned} \int _{r{\,\geqq \,}\rho }\omega ^2 \text {d}x\,\text {d}y=\int \limits _{\rho }^\infty dr\biggl (\int _{S_r} \omega ^2 ds\biggr ) {\,\leqq \,}2D_\lambda \end{aligned}$$

(with \(ds:=rd\theta \)), it is easy to show that there exists a \(\rho _* \in ((8\lambda )^{-1}, (4\lambda )^{-1})\) such that

$$\begin{aligned} \int _{S_{\rho _*}} \omega ^2 ds {\,\leqq \,}C\lambda D_\lambda ,\quad \quad -\left[ \partial _r \int _{S_{r}} \omega ^2 ds \right] _{r=\rho _*} {\,\leqq \,}C \lambda ^2 D_\lambda \end{aligned}$$
(3.29)

for some constant C. Then (3.29) and Lemma 14 together imply that

$$\begin{aligned} -\rho _* \int _{S_{\rho _*}} \partial _r \frac{\omega ^2}{2} ds {\,\leqq \,}C \lambda D_\lambda , \quad \quad \rho _* \int _{S_{\rho _*}} (\mathbf{w }\cdot \mathbf{e }_r )\frac{\omega ^2}{2} ds {\,\leqq \,}C \lambda D_\lambda . \end{aligned}$$
(3.30)

Hence, from (3.28) we deduced \(\int _{r {\,\geqq \,}\rho _*} r |\nabla \omega |^2 \,\text {d}x\text {d}y {\,\leqq \,}C \lambda D_\lambda \) for such \(\rho _*\). This proves the lemma. \(\square \)

Using (3.25) and Lemma 7 with \(\sigma =\frac{\lambda }{2}\), as well as the assumed convergence \(\mathbf{w } \rightarrow \lambda \mathbf{e }_1\) at infinity, we have

$$\begin{aligned} |\varphi (r)| {\,\leqq \,}\frac{C}{\lambda ^2} \int _{{\mathcal {E}} \backslash B_{(4\lambda )^{-1}}} \left( \frac{1}{r} |\nabla \omega | + |\nabla \mathbf{w }|^2 \right) \,\text {d}x\text {d}y \end{aligned}$$

for any \(r {\,\geqq \,}\frac{1}{4\lambda }\). Here \(\varphi (r)\) is the angle of \(\bar{\mathbf{w }}(r)\). From Lemma  15, and using Hölder’s inequality, we obtain

$$\begin{aligned} |\varphi (r)|&{\,\leqq \,}\frac{C}{\lambda ^2} \int _{{\mathcal {E}}\backslash B_{(4\lambda )^{-1}}} \biggl (\frac{\mu }{r^3} + \mu ^{-1}r|\nabla \omega |^2 + |\nabla \mathbf{w }|^2\biggr )\\&{\,\leqq \,}C\biggl (\frac{\mu }{\lambda }+\frac{D_\lambda }{\mu \lambda }+\frac{D_\lambda }{\lambda ^2}\biggr ) \\&{\,\leqq \,}C\frac{D_\lambda ^{\frac{1}{2}}}{\lambda } \end{aligned}$$

for any \(r {\,\geqq \,}\frac{1}{4\lambda }\) (taking \(\mu =D_\lambda ^{\frac{1}{2}}\) in the penultimate inequality). Together with (3.24), this implies that

$$\begin{aligned} |\bar{\mathbf{w }}(r) - \lambda \mathbf{e }_1| {\,\leqq \,}CD_\lambda ^{\frac{1}{2}} \end{aligned}$$
(3.31)

for all \(r {\,\geqq \,}\frac{1}{4\lambda }\). By (3.22), we immediately have that

$$\begin{aligned} |\mathbf{w } - \lambda \mathbf{e }_1| {\,\leqq \,}C D_\lambda ^{\frac{1}{2}} \end{aligned}$$
(3.32)

on \(S_{R_n}\) for \(n=-2, -1, \cdots \). These estimates on the good circles \(S_{R_n}\) will play important roles in the next two sections.

4 Pointwise estimates inside the critical circle (Stokes regime)

Let \(\mathbf{w }\) be an arbitrary solution to (\(1.1_\lambda \)) with \(0< \lambda < \min \{\lambda _2, \frac{1}{16}\}\) so that all the bounds in Section 3 are valid. In the present section the pointwise upper estimates for \(|\mathbf{w } - \lambda \mathbf{e }_1|\) will be derived within the bounded region \(\{r{\,\leqq \,}\lambda ^{-1}\} \cap {\mathcal {E}}\). They imply, in particular, that the required crucial estimates (1.14) are valid in this region.

We prove the following:

Lemma 16

The inequality \(|\mathbf{w }(z) - \lambda \mathbf{e }_1| {\,\leqq \,}C D_\lambda ^{\frac{1}{2}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{1}{2}}\) holds for all \(z \in \{r{\,\leqq \,}\lambda ^{-1}\} \cap {\mathcal {E}}\).

The method here is quite standard. It is based on the inequality (3.31) and accurate direct applications of linear Stokes estimates introduced in Section 2.3. The proof on Lemma 16 contains no surprising methods or ideas, and the reader can omit it on the first reading. So we moved this proof to the Appendix II.

The combination of Lemmas 16, 14 immediately implies

Corollary 17

\(|\mathbf{w }| {\,\leqq \,}C \lambda \) throughout \({\mathcal {E}}\).

5 Pointwise Estimates Outside the Critical Circle (Oseen Regime)

We assume \(0< \lambda < \min \{\lambda _2, \frac{1}{16}\}\) so that the estimates in Sections 3, 4 are valid. In this section, we prove the required decay estimates (1.14) for \(|\mathbf{w } - \lambda \mathbf{e }_1|\) in the unbounded region \(r > rsim \lambda ^{-1}\).

To make some notations simpler, we define a rescaled Navier–Stokes solution \(\mathbf{u }(z) = \lambda ^{-1} \mathbf{w }(\lambda ^{-1} z)\), \(q(z) = \lambda ^{-2} p(\lambda ^{-1} z)\) in the rescaled domain \(\lambda {\mathcal {E}}\). Let

$$\begin{aligned} \mathbf{v }=\mathbf{u }-\mathbf{e }_1\qquad \text{ and } \qquad {\varepsilon }= |\log \lambda |^{-\frac{1}{2}}. \end{aligned}$$

As usual, the components of \(\mathbf{u }, \mathbf{v }\) will be denoted by \(u_i, v_i\), \(i=1,2\).

Lemma 18

There exists a constant \(0< \lambda _3 < \min \{\lambda _2, \frac{1}{16}\}\), such that when \(0<\lambda <\lambda _3\), there holds \(|v_i(z)| {\,\leqq \,}C {\varepsilon }h_i(z)\) in \(|z| {\,\geqq \,}1, i=1,2\).

Here the majorant functions \(h_i(\xi )\) are the same as in Theorem 1, that is,

$$\begin{aligned} |\xi |>1 : \quad {\left\{ \begin{array}{ll} h_1(\xi ) = |\xi |^{-\frac{1}{2}}\\ h_2(\xi ) = |\xi |^{-\frac{3}{4}}. \end{array}\right. } \end{aligned}$$

Proof

With slight abuse of notation, in this proof, we still denote the Bernoulli function for \(\mathbf{u }\) by \(\Phi = \frac{|\mathbf{u }^2|}{2} + q\), and denote the vorticity by \(\omega = \partial _2 u_1 - \partial _1 u_2\). Hence \(\Phi \) and \(\omega \) here are different from those in previous sections. Define the stream function \(\psi \) for \(\mathbf{u }\) in \(\lambda {\mathcal {E}}\) by the relation \(\nabla \psi = \mathbf{u }^\perp =(-u_2,u_1)\). In particular, we have

$$\begin{aligned} \omega =\Delta \psi . \end{aligned}$$

For definiteness, put \(\psi (1,0) = 0\).

Let

$$\begin{aligned} \gamma = \Phi - \omega \psi . \end{aligned}$$

It is known that \(\gamma \) satisfies the two-sided maximum principle [1]. Now we divide the proof into a few steps.

Step 1: preparations. For convenience, in this step we collect and list some crucial information that we know on \(\mathbf{u }\) in the region \(\{r {\,\geqq \,}\frac{1}{4}\}\). By (3.1) we have

$$\begin{aligned} \int _{\lambda {{\mathcal {E}}}} |\nabla \mathbf{u }|^2 \text {d}x \text {d}y {\,\leqq \,}C {\varepsilon }^2. \end{aligned}$$
(5.1)

Further more, by Lemma 15 we obtain

$$\begin{aligned} \int _{r {\,\geqq \,}\frac{1}{4}} r|\nabla \omega |^2 \text {d}x\text {d}y {\,\leqq \,}C{\varepsilon }^2. \end{aligned}$$
(5.2)

As a consequence \(\int _{S_{r_1}} |\partial _\theta \omega |^2 {\,\leqq \,}C {\varepsilon }^2\) for some \(\frac{1}{4}< r_1 < \frac{1}{2}\). Then by virtue of Newton–Leibniz formula, since \(\omega \) changes sign on any circle surrounding the origin (see [1]), we get \(|\omega | {\,\leqq \,}C {\varepsilon }\) on \(S_{r_1}\). Using convergence \(\omega \rightarrow 0\) at infinity and the two-sided maximum principle again, we have \(|\omega | {\,\leqq \,}C {\varepsilon }\) in \(r {\,\geqq \,}r_1\). By Lemma 12, the pressure satisfies the inequality

$$\begin{aligned} |q| {\,\leqq \,}C{\varepsilon }\end{aligned}$$
(5.3)

in \(r {\,\geqq \,}\frac{1}{8}\). By the discussion on good circles \(S_{R_n}\) in Section 3, in particular (3.32), we have a sequence of radii \(r_n = \lambda R_n \in [2^n, 2^{n+1}), \ n= -2, -1, \cdots , 1,2, \cdots \) such that

$$\begin{aligned} |\mathbf{u } - \mathbf{e }_1| {\,\leqq \,}C {\varepsilon }\quad \text{ on } \text{ every } \, S_{r_n}. \end{aligned}$$
(5.4)

Combining (5.3) and (5.4), we get \(|\Phi - \frac{1}{2}| {\,\leqq \,}C{\varepsilon }\) on \(S_{r_{-2}}\). By Corollary 17, we have

$$\begin{aligned} |\mathbf{u }| {\,\leqq \,}C \end{aligned}$$
(5.5)

in \(\lambda {\mathcal {E}}\). Due to the definition of \(\psi \), we have also \(|\psi | {\,\leqq \,}C\) in \(\Omega _{\frac{1}{4}, 4}\). Since \(\gamma = \Phi - \omega \psi \), on \(S_{r_{-2}}\) we obtain \(|\gamma - \frac{1}{2}| {\,\leqq \,}|\Phi - \frac{1}{2}| + |\omega \psi | {\,\leqq \,}C{\varepsilon }\). It is proved in [1, Theorem 14(b)] that \(\gamma \rightarrow \frac{1}{2}\). By the two-sided maximum principle of \(\gamma \), we find that

$$\begin{aligned} |\gamma - \frac{1}{2}| {\,\leqq \,}C{\varepsilon }\end{aligned}$$
(5.6)

for any \(r {\,\geqq \,}r_{-2}\). We also point out here that by (5.1), (5.5) and the local regularity theory of Lemma 8, we have the pointwise control on derivatives

$$\begin{aligned} |\nabla \mathbf{u }|, |\nabla ^2 \mathbf{u }| {\,\leqq \,}C{\varepsilon }\end{aligned}$$
(5.7)

in \(r {\,\geqq \,}\frac{1}{4}\).

Step 2: pointwise smallness of \(\mathbf{v }\). In \({\mathcal {E}}_1\), we claim that

$$\begin{aligned} |\mathbf{v }| {\,\leqq \,}C{\varepsilon }. \end{aligned}$$
(5.8)

The proof of this claim is essentially based on the classical works [1, 12], and on the recent work [18]. In [1], Amick proved that the absolute value \(|\mathbf{u }|\) is close to the limiting value 1 using the smallness of three quantities: the Dirichlet energy, \(|\gamma - \frac{1}{2}|\), and the pressure. We refer to the proof of Theorem 21(a) in [1] for details. In our situation, such smallness is evidently provided by (5.1)–(5.6). Further, in [18, Lemma 3.3(ii)] Korobkov et al. proved the smallness of \(|\mathbf{v }|=|\mathbf{u }-\mathbf{e }_1|\) using Amick’s result and the additional observation that the angle of velocity is also under control.

In our situation, we have to modify some of the arguments in [1] so that the smallness factor \({\varepsilon }\) can be fully preserved. The complete proof of (5.8) is presented in our Appendix I. In fact, many technical moments of Amick’s proof can be simplified using [18, Remark 4.1].

Step 3: summability of \(\mathbf{v }\). For this step, we mainly use the potential estimates for Oseen system which are developed in [27, Lemmas 1–2] by L.I.Sazonov (see also [9, 28] ). As will be shown, the pointwise smallness of \(\mathbf{v }\) and the smallness of Dirichlet energy together are sufficient to control certain Lebesgue norms of \(\mathbf{v }\) in the exterior domain by the regularity estimates near the boundary.

We write \({\mathcal {E}}_\rho = \{r {\,\geqq \,}\rho \}\) and write \(\mathbf{E }_i = (E_{1i}, E_{2i}), \ i = 1, 2\), where \(E_{ij}\) is the Oseen tensor introduced in Section 2.4. Using (2.15) in the domain \({\mathcal {E}}_1\) and applying suitable integration by parts (and using \(\partial _1 v_1 + \partial _2 v_2 = 0\)), we obtain

$$\begin{aligned} v_2(z)&= \int _{{\mathcal {E}}_1} (v_1 \partial _2 E_{12} - v_1 \partial _1 E_{22} - v_2 \partial _2 E_{22} + 2\partial _2 v_1 \, E_{12}) v_2 \nonumber \\&\quad -\, \int _{S_1} (v_1^2 n_1 + 2v_1v_2n_2) E_{12} -\int _{S_1} \mathbf{v }(z') \cdot \mathbf{T }_z(\mathbf{E }_2, e_2)(z-z') \cdot \mathbf{n } \nonumber \\&\quad -\, \int _{S_1} \mathbf{E }_2(z-z') \cdot \mathbf{T }(\mathbf{v },q)(z') \cdot \mathbf{n } + \int _{S_1} (\mathbf{v }(z') \cdot \mathbf{E }_2(z-z')) (\mathbf{n } \cdot \mathbf{u }(z')) \nonumber \\&=: A_2(z) + B_2(z). \end{aligned}$$
(5.9)

Here \(A_2\) and \(B_2\) stand for the area integrals and the boundary integrals respectively. The area integrals are taken over the variable \(z'\). Note that \(E_{ij}\) appearing in the integrals should be understood as \(E_{ij}(z-z')\) and the derivatives on \(E_{ij}\) act on the variable \(z'\). For representation of \(v_1\), we simply use (2.15) without integration by parts to get

$$\begin{aligned} v_1(z)&= - \int _{{\mathcal {E}}_1} (v_1 \partial _1 E_{11} + v_2 \partial _1 E_{21} + v_2 \partial _2 E_{11}) v_1 + v_2^2 \partial _2 E_{21} \nonumber \\&\quad -\,\int _{S_1} \mathbf{v }(z') \cdot \mathbf{T }_z(\mathbf{E }_1, e_1)(z-z') \cdot \mathbf{n } \nonumber \\&\quad -\, \int _{S_1} \mathbf{E }_1(z-z') \cdot \mathbf{T }(\mathbf{v },q)(z') \cdot \mathbf{n } + \int _{S_1} (\mathbf{v }(z') \cdot \mathbf{E }_1(z-z')) (\mathbf{n } \cdot \mathbf{u }(z')) \nonumber \\&=: A_1(z) + B_1(z). \end{aligned}$$
(5.10)

As before, \(A_1\) stands for the area integrals and \(B_1\) stands for the boundary integrals.

Of course, the key issue here is to estimate the area integrals, because decay of boundary integrals outside the unit disk can be estimated relatively easily (using the uniform smallness of \(\mathbf{v }\)).

Let us recall some estimates of \(\mathbf{E }\). By the exact form of \(\mathbf{E }\) and the asymptotic of \(K_0\) at infinity (see Section 2.4), we have

$$\begin{aligned}&\displaystyle E_{11} \in L^{3, \infty } \cap L^{3+\delta },\\&\displaystyle E_{12} = E_{21}, E_{22} \in L^{2, \infty } \cap L^{2+\delta } \end{aligned}$$

in \({\mathbb {R}}^2\) for any finite \(\delta > 0\), and

$$\begin{aligned} \partial _2 E_{11} \in L^{\frac{3}{2},\infty } \end{aligned}$$

in \({\mathbb {R}}^2\). Here \(L^{s,\infty }\) is the weak \(L^s\)-space. Moreover, by the Fourier transform (2.11) of \(\mathbf{E }\) and by the Mikhlin multiplier theorem (see, e.g., [30, Chapter VI,§4–5]) we have

$$\begin{aligned} \Vert \partial _k E_{ij} * f\Vert _{L^s({\mathbb {R}}^2)} {\,\leqq \,}C_s\,\Vert f\Vert _{L^s({\mathbb {R}}^2)} \end{aligned}$$

for any \(f\in L^s({\mathbb {R}}^2)\), \(1<s<\infty \), \((i,j,k) \ne (1,1,2)\).Footnote 8 It is also known that \(v_1 \in L^{3+\delta }({\mathcal {E}}_1)\) and \(v_2 \in L^{2+\delta }({\mathcal {E}}_1)\) for any \(\delta > 0\) by the estimates (1.15). With the above bounds, using weak Young inequality for convolutions (see, e.g., [22, Section 4.3]), we deduce from (5.9) that

$$\begin{aligned} \Vert v_2\Vert _{L^s({\mathcal {E}}_1)}&{\,\leqq \,}\Vert A_2\Vert _{L^s({\mathcal {E}}_1)} + \Vert B_2\Vert _{L^s(\Omega _{1,2})} + \Vert B_2\Vert _{L^s({\mathcal {E}}_2)} \nonumber \\&{\,\leqq \,}2\Vert A_2\Vert _{L^s({\mathcal {E}}_1)} + \Vert v_2\Vert _{L^s(\Omega _{1,2})} + \Vert B_2\Vert _{L^s({\mathcal {E}}_2)} \nonumber \\&{\,\leqq \,}C_s \left( \Vert \mathbf{v }\Vert _{L^\infty ({\mathcal {E}}_1)} + \Vert \partial _2 v_1\Vert _{L^2({\mathcal {E}}_1)} \right) \Vert v_2\Vert _{L^s({\mathcal {E}}_1)} \nonumber \\&\quad +\, \Vert v_2\Vert _{L^s(\Omega _{1,2})} + \Vert B_2\Vert _{L^s({\mathcal {E}}_2)} \end{aligned}$$
(5.11)

for any \(2< s < \infty \). Using information from Step 1 and Step 2, we have

$$\begin{aligned} \Vert \mathbf{v }\Vert _{L^\infty ({\mathcal {E}}_1)} + \Vert \partial _2 v_1\Vert _{L^2({\mathcal {E}}_1)} {\,\leqq \,}C {\varepsilon }\end{aligned}$$

and

$$\begin{aligned} \Vert v_2\Vert _{L^s(\Omega _{1,2})} + \Vert B_2\Vert _{L^s({\mathcal {E}}_2)} {\,\leqq \,}C \Vert \mathbf{v }\Vert _{C^1(\Omega _{\frac{1}{2},2})} + \Vert q\Vert _{L^\infty (\Omega _{\frac{1}{2},2})} {\,\leqq \,}C_s {\varepsilon }\end{aligned}$$

for any \(2< s < \infty \). Hence, when \({\varepsilon }\) is sufficiently small (depending on the choice of s), or equivalently, when \(\lambda \) is sufficiently small, we obtain from (5.11) that

$$\begin{aligned} \Vert v_2\Vert _{L^s({\mathcal {E}}_1)} {\,\leqq \,}C_s{\varepsilon }\end{aligned}$$
(5.12)

for any \(2< s < \infty \). Similarly, using (5.10), we have

$$\begin{aligned} \Vert v_1\Vert _{L^m({\mathcal {E}}_1)}&{\,\leqq \,}\Vert A_1\Vert _{L^m({\mathcal {E}}_1)} + \Vert B_1\Vert _{L^m(\Omega _{1,2})} + \Vert B_1\Vert _{L^m({\mathcal {E}}_2)} \nonumber \\&{\,\leqq \,}2\Vert A_1\Vert _{L^m({\mathcal {E}}_1)} + \Vert v_1\Vert _{L^m(\Omega _{1,2})} + \Vert B_1\Vert _{L^m({\mathcal {E}}_2)} \nonumber \\&{\,\leqq \,}C_m \left( \Vert \mathbf{v }\Vert _{L^\infty ({\mathcal {E}}_1)} + \Vert v_2\Vert _{L^3({\mathcal {E}}_1)} \right) \Vert v_1\Vert _{L^m({\mathcal {E}}_1)} + C_m\Vert v_2\Vert _{L^{2m}({\mathcal {E}}_1)}^2 \nonumber \\&\quad +\, \Vert v_1\Vert _{L^m(\Omega _{1,2})} + \Vert B_2\Vert _{L^m({\mathcal {E}}_2)} \end{aligned}$$
(5.13)

for any \(3< m < \infty \). Using (5.12) and similar arguments as those for \(v_2\), we deduce

$$\begin{aligned} \Vert v_1\Vert _{L^m({\mathcal {E}}_1)} {\,\leqq \,}C_m{\varepsilon }\end{aligned}$$
(5.14)

for any \(3< m < \infty \), when \(\lambda \) is sufficiently small (depending on the choice of m).

Step 4: pointwise decay of \(\mathbf{v }\). First, we prove a pointwise decay estimate for vorticity using an idea of Gilbarg and Weinberger [12]. By Hölder’s inequality,

$$\begin{aligned} \int _1^\infty \frac{dr}{r} \int _0^{2\pi } |\partial _\theta ( r^\frac{3}{2} \omega ^2)| \text {d}\theta&= 2\int _{{\mathcal {E}}_1} r^{-\frac{1}{2}} |\omega \partial _\theta \omega | \text {d}x\, \text {d}y \\&{\,\leqq \,}\int _{{\mathcal {E}}_1} \omega ^2 \text {d}x\, \text {d}y + \int _{{\mathcal {E}}_1} r |\nabla \omega |^2 \text {d}x \, \text {d}y \\&{\,\leqq \,}C{\varepsilon }^2. \end{aligned}$$

Hence, for each \(n = 0,1,2, \cdots \), there exists \(r \in [2^n, 2^{n+1})\) such that

$$\begin{aligned} \int _0^{2\pi } |\partial _\theta ( r^\frac{3}{2} \omega ^2)| \text {d}\theta {\,\leqq \,}C{\varepsilon }^2. \end{aligned}$$

Recall that there are two curves \(\lambda {\mathcal {C}}_i, i=1,2\) with \({\mathcal {C}}_i\) given by Lemma 13 such that \(\omega \) vanishes on them. Evidently, \(S_r\) intersects \(\lambda {\mathcal {C}}_i\) for any \(r {\,\geqq \,}1\). Hence, for \(r \in [2^n, 2^{n+1})\) given above, we have

$$\begin{aligned} r^\frac{3}{4} \max _{S_r}|\omega | {\,\leqq \,}C{\varepsilon }. \end{aligned}$$
(5.15)

By the two-sided maximum principle for \(\omega \), the above estimate holds for any \(r {\,\geqq \,}2\).

For any disk \(B_\rho (z)\subset {{\mathcal {E}}}\), the following standard identity holds:

$$\begin{aligned} \mathbf{v }(z)=\frac{1}{2\pi }\int \limits _{\partial B_\rho (z)}\frac{\mathbf{v }(\zeta )}{\rho }\,ds - \frac{1}{2\pi }\int \limits _{B_\rho (z)}\frac{\omega (\zeta )(z-\zeta )^\bot }{|z-\zeta |^2}\,\text {d}\zeta . \end{aligned}$$

Here \((z-\zeta )^\bot =(-(z_2-\zeta _2),z_1-\zeta _1)\in {{\mathbb {R}}}^2\). By virtue of this identity, following [27, Section 5], using \(\Vert \mathbf{v }\Vert _{L^m(r {\,\geqq \,}1)} {\,\leqq \,}C_m {\varepsilon }, \ m>3\) from Step 3, and (5.15), we immediately reach the pointwise bound

$$\begin{aligned} |\mathbf{v }(z)| {\,\leqq \,}C_\delta {\varepsilon }\, r^{-\frac{3}{10} + \delta } \end{aligned}$$
(5.16)

in \({\mathcal {E}}_1\) for any \(\delta > 0\) (when \(\lambda \) is sufficiently small). Choose and fix a small \(\delta \) such that \(\frac{3}{10} - \delta > \frac{1}{4}\). Now the meaning of “\(\lambda \) being sufficiently small” in Step 3 is also fixed.

Next, we use the representation formula (2.15) in \({\mathcal {E}}_1\) again to get

$$\begin{aligned} \mathbf{v }(z)&= - \int _{{\mathcal {E}}_1} (\mathbf{v } \cdot \nabla _{z'}) \mathbf{E }(z - z') \cdot \mathbf{v } \nonumber \\&-\int _{S_1} \mathbf{v }(z') \cdot \mathbf{T }_z(\mathbf{E }, \mathbf{e })(z-z') \cdot \mathbf{n } \nonumber \\&- \int _{S_1} \mathbf{E }(z-z') \cdot \mathbf{T }(\mathbf{v },p)(z') \cdot \mathbf{n } + \int _{S_1} (\mathbf{v }(z') \cdot \mathbf{E }(z-z')) (\mathbf{n } \cdot \mathbf{u }(z')) \nonumber \\&=: \mathbf{N } + \mathbf{L }. \end{aligned}$$
(5.17)

Here \(\mathbf{L }\) are the sum of all boundary integrals and \(\mathbf{N }\) is the area integral. By the asymptotic form of \(\mathbf{E }\), we have

$$\begin{aligned} |L_i| {\,\leqq \,}C {\varepsilon }h_i(z) \end{aligned}$$
(5.18)

in \(r {\,\geqq \,}2\). It is easy to check that \(|\mathbf{N }| {\,\leqq \,}C {\varepsilon }^2\) in \(\Omega _{\frac{1}{2},2}\). Hence \(|\mathbf{L }| {\,\leqq \,}|\mathbf{v }| + |\mathbf{N }| {\,\leqq \,}C{\varepsilon }\) in \(\Omega _{1,2}\), and as a consequence, (5.18) holds in \(r {\,\geqq \,}1\).

Now it remains to estimate the second term \(\mathbf{N }\) in (5.17) (area integrals). This can be done using Lemmas 1 and 2 from the classical Smith paper [29, p.361]. These lemmas give some self-improving estimates for term \(\mathbf{N }\), that is, if we assume a priori that \(\mathbf{v }\) has the uniform decay of type (5.16), then a posteriori \(\mathbf{N }\) has better decay, etc. So we can use these two lemmas (with parameter \(\sigma = 0\) there) finitely many times to improve the bound (5.16) until \(\mathbf{N }\) is shown to be decaying faster than the right of (5.18). This concludes the proof of Lemma 18. \(\square \)

Now we are ready to give the

Proof of Lemma 3

Lemma 18 implies that, when \(\lambda \) is sufficiently small,

$$\begin{aligned} |(\mathbf{w }(z)-\lambda \mathbf{e }_1)_i(z)| {\,\leqq \,}C |\log \lambda |^{-\frac{1}{2}} \lambda h_i(\lambda z), i=1,2 \end{aligned}$$

in the exterior region \({\mathcal {E}}_{\lambda ^{-1}} = \{z : r{\,\geqq \,}\lambda ^{-1}\}\). By Lemmas 16 and 9, the above also holds true in the bounded region \(\{z : r {\,\leqq \,}\lambda ^{-1}\} \cap {\mathcal {E}}\) (with a different positive constant C). Hence, (1.14) holds throughout \({\mathcal {E}}\) for some positive constant \(M_1\). It remains to take \(\lambda \) sufficiently small so that \(M_1 |\log \lambda |^{-\frac{1}{2}} {\,\leqq \,}{\varepsilon }_0\) where \({\varepsilon }_0\) is given by Theorem 1.

\(\square \)

Now the assertion of the main Theorem 2 follows immediately from Lemma 3 and from the conditional uniqueness result in Finn–Smith Theorem 1. The proof is finished.