Uniqueness of Plane Stationary Navier–Stokes Flow Past an Obstacle

Abstract

We study the exterior problem for stationary Navier–Stokes equations in two dimensions describing a viscous incompressible fluid flowing past an obstacle. It is shown that, at small Reynolds numbers, the classical solutions constructed by Finn and Smith are unique in the class of D-solutions (that is, solutions with finite Dirichlet integral). No additional symmetry or decay assumptions are required. This result answers a long-standing open problem. In the proofs, we developed the ideas of the classical Ch. Amick paper (Acta Math. 1988).

Appendices

Appendix I

Proof of Step 2

In this section we discuss the proof of the uniform pointwise estimate (5.8), that is,

$$\begin{aligned} |\mathbf{u }-(1,0)|=|{\mathbf{v }}|{\,\leqq \,}C{\varepsilon }\end{aligned}$$

for $r{\,\geqq \,}1.$

Step 2a. First of all, consider the “good” cone

$$\begin{aligned} {\mathcal {K}}^{\pm y}=\biggl \{r{\,\geqq \,}1,|\theta |\in \bigl (\frac{1}{5}\pi ,\frac{4}{5}\pi \bigr )\biggr \}, \end{aligned}$$

which is separated from the x-axis. Here one can simply follow the proof of [1, Theorem 19]. The key observation is that $|\psi |{\,\geqq \,}cr>0$ in such a cone, and this fact, by virtue of smallness, gives

$$\begin{aligned} |\Phi -\frac{1}{2}|{\,\leqq \,}C{\varepsilon },\qquad |\gamma -\frac{1}{2}|=|\Phi -\frac{1}{2}-\omega \psi |{\,\leqq \,}C{\varepsilon }\end{aligned}$$

on good circles and by maximum principle for $\omega$, implies $|\omega | {\,\leqq \,}C{\varepsilon }r^{-1}$ in ${\mathcal {K}}^{\pm y}$, which easily gives the required smallness $|\mathbf{u }-(1,0)|{\,\leqq \,}C{\varepsilon }$ here.

Now consider the more complicated region

$$\begin{aligned} \bigl \{r{\,\geqq \,}1,|\theta |{\,\leqq \,}\frac{1}{5}\pi \bigr \}. \end{aligned}$$

Here the previous estimate $|\omega | {\,\leqq \,}C{\varepsilon }r^{-1}$ does not hold in general, so the arguments should be more delicate and subtle. The main ideas here are due to Amick [1] and Korobkov et al. [18, Remark 4.1].

Step 2b. We are going to use the uniform smallness of $\gamma$-function

$$\begin{aligned} \bigl |\gamma -\frac{1}{2}\bigr |=\bigl |\Phi -\frac{1}{2}-\omega \psi \bigr |{\,\leqq \,}C{\varepsilon }\end{aligned}$$

and the level sets of the stream function $\psi$. On the first step here we show that the set ${\mathcal {C}} = \{r{\,\geqq \,}\frac{1}{2}, \psi = 0\}$ consists of exactly two smooth curves ${\mathcal {C}}_{\pm }$. Indeed, from Step 1, we know that

$$\begin{aligned} |\psi - y| {\,\leqq \,}C{\varepsilon }\end{aligned}$$

(A.1)

in $\Omega _{\frac{1}{2},2}$. Using (5.1), there exists an angle $\theta _1 \in [\frac{\pi }{10}, \frac{\pi }{5})$ such that

$$\begin{aligned} \int \limits _{1/4}^{+\infty } |\partial _r \mathbf{u }(r, \theta )|^2 rdr {\,\leqq \,}C{\varepsilon }^2 \end{aligned}$$

for $\theta = \theta _1, -\theta _1, \pi - \theta _1, -\pi + \theta _1$. Hence, for such $\theta$,

$$\begin{aligned} \int _{r_n}^{r_{n+1}} |\partial _r \mathbf{u }(r, \theta )| dr&{\,\leqq \,}C {\varepsilon }\left( \int _{r_n}^{r_{n+1}} \frac{dr}{r}\right) ^\frac{1}{2}, \\&{\,\leqq \,}C{\varepsilon }\end{aligned}$$

with $r_n, n=-2,-1,\cdots$ given in Step 1. In view of (5.4), we obtain

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

for any $r{\,\geqq \,}\frac{1}{2}$ and $\theta = \theta _1, -\theta _1, \pi - \theta _1, -\pi + \theta _1$. By the definition of $\psi$ and (A.1), we have

$$\begin{aligned} |\psi - y| {\,\leqq \,}C{\varepsilon }r \end{aligned}$$

(A.2)

pointwise on the set

$$\begin{aligned} {\mathcal {N}} := \bigl (\cup _{n {\,\geqq \,}-2} S_{r_n}\bigr )\cup \bigl \{r {\,\geqq \,}\frac{1}{2}, \theta = \theta _1, -\theta _1, \pi - \theta _1, -\pi + \theta _1\bigr \}. \end{aligned}$$

(A.3)

When $\lambda$ is small, this implies that ${\mathcal {C}}$ must intersect ${\mathcal {N}}$ within the cone ${\mathcal {K}} := \{r{\,\geqq \,}\frac{1}{2}, |\theta |< \frac{\pi }{10} \ \text {or}\ |\pi - \theta | < \frac{\pi }{10}\}$. Moreover, by (5.4), ${\mathcal {C}}$ intersects each $S_{r_n}, n{\,\geqq \,}-2$ at exactly two points, one with $x>0$ and another with $x<0$. On the level set ${\mathcal {C}}$ we have $\Big |\frac{|\mathbf{u }|^2}{2} - \frac{1}{2}\Big | = |\gamma - \frac{1}{2} - q| {\,\leqq \,}C{\varepsilon }$ (see (5.3), (5.6) ), hence

$$\begin{aligned} \Big ||\mathbf{u }| - 1\Big | {\,\leqq \,}C{\varepsilon }\quad \text{ on } {\mathcal {C}}. \end{aligned}$$

(A.4)

As a consequence, $|\nabla \psi |=|{\mathbf{u }}|\ne 0$ on the set ${\mathcal {C}}$, that is, ${\mathcal {C}}$ is a regular curve in the case when $\lambda$ is small.

Further, when $\lambda$ is small, ${\mathcal {C}}$ cannot contain any closed curve ${\mathcal {L}}$ by an elegent idea of Amick’s. To explain this, suppose ${\mathcal {C}}$ contains a closed curve ${\mathcal {L}}$. Let ${\mathcal {U}}$ denote the domain bounded by ${\mathcal {L}}$, then

$$\begin{aligned} \left( \int _{\mathcal {U}} \omega ^2\, \text {d}x\text {d}y\right) ^\frac{1}{2} |{\mathcal {U}}|^\frac{1}{2} {\,\geqq \,}\left| \int _{\mathcal {U}} \omega \, \text {d}x\text {d}y\right| = \left| \int _{\mathcal {U}} \Delta \psi \, \text {d}x\text {d}y\right| = \left| \int _{\mathcal {L}} \partial _n \psi \,ds\right| = \\ =\int _{\mathcal {L}} |\nabla \psi |\,ds=\int _{\mathcal {L}} |{\mathbf{u }}|\,ds\overset{(A.4)}{{\,\geqq \,}}\frac{1}{2}|{\mathcal {L}}|. \end{aligned}$$

(Note, that on ${\mathcal {L}}$ one of the identities $\partial _n \psi \equiv |\nabla \psi |$ or $\partial _n \psi \equiv -|\nabla \psi |$ holds, because ${\mathcal {L}}$ is regular closed level set of $\psi$.) This implies

$$\begin{aligned} |{\mathcal {L}}| {\,\leqq \,}C {\varepsilon }|{\mathcal {U}}|^\frac{1}{2}, \end{aligned}$$

which contradicts with the isoperimetric inequality when ${\varepsilon }$ is small. Hence ${\mathcal {C}}$ must consist of two smooth curves starting from $r=\frac{1}{2}$ and extending to infinity, each contained in ${\mathcal {K}}\cap \{x>0\}$ and ${\mathcal {K}}\cap \{x<0\}$ respectively. We denote these two curves by ${\mathcal {C}}_{\pm }$.

Step 2c. Here we show that

$$\begin{aligned} |\mathbf{v }| {\,\leqq \,}C{\varepsilon }\quad \text{ along } {\mathcal {C}}_\pm . \end{aligned}$$

(A.5)

Take any point $z \in {\mathcal {C}}_+$ (${\mathcal {C}}_-$ would be similar) with $r=|z| {\,\geqq \,}1$. By Lemma 6, there exist good circles $S_{{\tilde{r}}_n}$ centered at z with ${\tilde{r}}_n \in [2^{-n-1}r, 2^{-n}r), n=1,2, \cdots$, such that

$$\begin{aligned} |\mathbf{u } - \bar{\mathbf{u }}^{(n)}| {\,\leqq \,}C{\varepsilon }\quad \text{ on } S_{{\tilde{r}}_n}. \end{aligned}$$

Here $\bar{\mathbf{u }}^{(n)}$ is the average of $\mathbf{u }$ on $S_{{\tilde{r}}_n}$. Since $S_{{\tilde{r}}_n}$ must intersect ${\mathcal {C}}_+$ and on ${\mathcal {C}}_+$ the inequality (A.4) holds, we obtain

$$\begin{aligned} \left| |\bar{\mathbf{u }}^{(n)}| - 1\right| {\,\leqq \,}C{\varepsilon }\end{aligned}$$

on each $S_{{\tilde{r}}_n}$. From this, (2.1) of Lemma 6 implies

$$\begin{aligned} \left| |\bar{\mathbf{u }}^{(\rho )}| - 1\right| {\,\leqq \,}C{\varepsilon }, \end{aligned}$$

where $\bar{\mathbf{u }}^{(\rho )}$ is the mean value of $\mathbf{u }$ on circles $S_\rho (z)$ centered at z with radius $\rho {\,\leqq \,}\frac{r}{2}$. Let $\varphi ^{(n)}$ be the angle of $\bar{\mathbf{u }}^{(n)}$. By Lemma 7, we have, for any $n,m{\,\geqq \,}1$,

$$\begin{aligned} |\varphi ^{(n)} - \varphi ^{(m)}|&{\,\leqq \,}C \int _{|\zeta -z| {\,\leqq \,}\frac{r}{2}} \frac{|\nabla \omega |}{|\zeta -z|} + |\nabla \mathbf{u }|^2 \, \text {d}\zeta _1 \text {d}\zeta _2 \nonumber \\&{\,\leqq \,}C \left( \int _{1{\,\leqq \,}|\zeta -z| {\,\leqq \,}\frac{r}{2}} \frac{|\nabla \omega |}{|\zeta -z|} \text {d}\zeta _1 \text {d}\zeta _2\right) \end{aligned}$$

(A.6)

$$\begin{aligned}&\quad +\, C \left( \int _{|\zeta -z| {\,\leqq \,}1} \frac{|\nabla \omega |}{|\zeta -z|} \text {d}\zeta _1 \text {d}\zeta _2\right) + C{\varepsilon }^2 \nonumber \\&{\,\leqq \,}C {\varepsilon }. \end{aligned}$$

(A.7)

In this last line we have used (5.2) and (5.7) for the first and second terms in the penultimate inequality respectively. By construction, the circle $S_{{\tilde{r}}_1}$ must intersect the set ${\mathcal {N}}$ (see (A.3) ) on which $|\mathbf{v }| {\,\leqq \,}C{\varepsilon }$. Hence, $|\bar{\mathbf{u }}^{(1)} - \mathbf{e }_1| {\,\leqq \,}C{\varepsilon }$. Letting $n=1, m\rightarrow \infty$ in (A.6), and using (A.4), we get $|\mathbf{v }(z)| {\,\leqq \,}C{\varepsilon }$. Note that this implies that the slope of ${\mathcal {C}}_\pm$ is small.

Step 2d. Now take arbitrary point $z_1 = (x_1,y_1) \in \{r{\,\geqq \,}1, |\theta |<\frac{\pi }{5}, x>0\}$ (the case $x<0, |\pi - \theta |< \frac{\pi }{5}$ is similar) and show that

$$\begin{aligned} |\mathbf{v }(z_1)| {\,\leqq \,}C{\varepsilon }. \end{aligned}$$

Let $z_2 = (x_1,y_2) \in {\mathcal {C}}_+$. To simplify notations, let’s change the coordinate system, namely, let’s move the coordinate origin to the point $z_2$, so now

$$\begin{aligned} z_2=(x_1,y_2)=(0,0)\in {\mathcal {C}}_+, \quad z_1=(x_1,y_1)=(0,y_1). \end{aligned}$$

Consider the case $y_1 >0$, that is, when the point $z_1$ is above the ${\mathcal {C}}_+$ curve, so that $\psi (z_1) > 0$ (the opposite case $y_1<0$ case is quite similar). Let

$$\begin{aligned} R = y_1. \end{aligned}$$

Using Lemma 6, we find two good circles $S_{R_1}(z_1)$ and $S_{R_2}(z_2)$ centered at $z_1$ and $z_2$ respectively, with radii

$$\begin{aligned} R_1, R_2 \in \bigl (\frac{2R}{3},\frac{3R}{4}\bigr ). \end{aligned}$$

Clearly,

$$\begin{aligned} S_{R_1}(z_1)\cap S_{R_2}(z_2)\ne \emptyset \ne {\mathcal {C}}_+\cap S_{R_2}(z_2). \end{aligned}$$

As a result, on both $S_{R_1}(z_1)$ and $S_{R_2}(z_2)$ we have

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

(A.8)

Note that $\psi = 0$ on ${\mathcal {C}}_+$. We can integrate $\nabla (\psi -y) = \mathbf{v }^\perp$ starting from $z_2\in {\mathcal {C}}_+$ along arcs of ${\mathcal {C}}_+$, $S_{R_2}(z_2)$ and $S_{R_1}(z_1)$. This process gives $|\psi - y| {\,\leqq \,}C{\varepsilon }R$ on $S_{R_1}(z_1)$. As a consequence, by virtue of the evident inequalities

$$\begin{aligned} \frac{1}{4}R{\,\leqq \,}\min \limits _{(x,y)\in S_{R_1}(z_1)}y<\max \limits _{(x,y)\in S_{R_1}(z_1)}y<\frac{7}{4} R, \end{aligned}$$

we have

$$\begin{aligned} \left| \frac{\psi }{y} - 1\right| {\,\leqq \,}C{\varepsilon }\end{aligned}$$

(A.9)

on $S_{R_1}(z_1)=\partial B_{R_1}(z_1)$.

Now we are going to prove that the last estimate is valid not only for boundary circle $S_{R_1}(z_1)$, but for all points of the disk $B_{R_1}(z_1)$. Recall that

$$\begin{aligned} |\gamma - q -\frac{1}{2}| = \left| \frac{|\nabla \psi |^2}{2} - \psi \Delta \psi - \frac{1}{2} \right| {\,\leqq \,}C{\varepsilon }\end{aligned}$$

(A.10)

pointwisely holds in the region $\{r{\,\geqq \,}1\}$ (see (5.3), (5.6) ). Denote $\psi = y(1+S)$ in $B_{R_1}(z_1)$. Since ${\mathcal {C}}_+\cap B_{R_1}(z_1)=\emptyset$ and $\psi$ is positive on $B_{R_1}(z_1)$, by construction we have that the values $(1+S)$ are positive on $B_{R_1}(z_1)$ as well. Assume at the moment, that S has positive maximum at the interior point of the disk $B_{R_1}(z_1)$. Then at this maximum point $\nabla S=0$ and $\Delta S{\,\leqq \,}0$, therefore,

$$\begin{aligned} |\nabla \psi |^2 - 2\psi \Delta \psi = (1+S)^2-2\psi y\Delta S{\,\geqq \,}(1+S)^2, \end{aligned}$$

which, by virtue of (A.10), implies $(1+S)^2 {\,\leqq \,}1 + C{\varepsilon },$ and consequently,

$$\begin{aligned} S {\,\leqq \,}C{\varepsilon }\end{aligned}$$

at any maximum point of S inside the disk $B_{R_1}(z_1)$. Similarly, a consideration for the negative minimal points of S gives $S {\,\geqq \,}-C{\varepsilon }$. Hence, taking into account (A.9), we have proved

$$\begin{aligned} \left| \frac{\psi }{y} - 1\right| {\,\leqq \,}C{\varepsilon }\end{aligned}$$

(A.11)

in $B_{R_1}(z_1)$. In particular,

$$\begin{aligned} |\psi -y|{\,\leqq \,}C{\varepsilon }R\qquad \text{ in } \ B_{R_1}(z_1). \end{aligned}$$

Next, one observes that

$$\begin{aligned} \Delta (\sqrt{\psi } - \sqrt{y}) = \frac{2\psi \Delta \psi - |\nabla \psi |^2}{4 \psi ^\frac{3}{2}} + \frac{1}{4y^\frac{3}{2}}. \end{aligned}$$

Hence, using (A.10) and (A.11), we get

$$\begin{aligned} |\Delta (\sqrt{\psi } - \sqrt{y})| {\,\leqq \,}C{\varepsilon }y^{-\frac{3}{2}} {\,\leqq \,}C{\varepsilon }R^{-\frac{3}{2}} \end{aligned}$$

(A.12)

in the disc $B_{R_1}(z_1)$. On the other hand, (A.11) implies

$$\begin{aligned} |\sqrt{\psi } - \sqrt{y}| {\,\leqq \,}C{\varepsilon }R^{\frac{1}{2}} \end{aligned}$$

(A.13)

in $B_{R_1}(z_1)$. Now, applying the standard estimates for Laplac operator in the unit disk and scaling to the function $\sqrt{\psi } - \sqrt{y}$ with (A.12)–(A.13), we obtain

$$\begin{aligned} |\nabla (\sqrt{\psi } - \sqrt{y})| {\,\leqq \,}C{\varepsilon }R^{-\frac{1}{2}} \end{aligned}$$

inside $\frac{1}{2}B_{R_1}(z_1)$, that implies the required estimate $|\nabla \psi (z_1) - \mathbf{e }_1^\perp |=|\mathbf{u }-\mathbf{e }_1| {\,\leqq \,}C{\varepsilon }$, see [1, Proof of Theorem 27, page 118].

$\square$

Appendix II

Proof of Lemma 16

Let $w_r = \mathbf{w } \cdot \mathbf{e }_r$ be the radial component of $\mathbf{w }$. We need the following classical inequality:

$$\begin{aligned} \frac{d}{dr} \int _0^{2\pi } |\mathbf{w }(r, \theta ) - \bar{\mathbf{w }}(r)|^2 \text {d}\theta&= \int _0^{2\pi } 2w_r \cdot (\mathbf{w } - \bar{\mathbf{w }}(r)) \text {d}\theta \nonumber \\&{\,\leqq \,}\int _0^{2\pi } \left[ r|w_r|^2 + \frac{|\mathbf{w } - \bar{\mathbf{w }}(r)|^2}{r} \right] \text {d}\theta \nonumber \\&{\,\leqq \,}\int _0^{2\pi } |\nabla \mathbf{w }|^2 r\text {d}\theta . \end{aligned}$$

(B.1)

By our assumption on the domain ${\mathcal {E}}$, we have $\{r{\,\geqq \,}1\}\subset {\mathcal {E}}$. By integrating (B.1) on the interval $[r, \infty )$, and using the fact that $\mathbf{w } \rightarrow \lambda \mathbf{e }_1$ uniformly at infinity, we obtain

$$\begin{aligned} \int _0^{2\pi } |\mathbf{w }(r, \theta ) - \bar{\mathbf{w }}(r)|^2 \text {d}\theta {\,\leqq \,}D_\lambda \end{aligned}$$

(B.2)

for any $r{\,\geqq \,}1$. By (2.1) and (3.31), for $1 {\,\leqq \,}r {\,\leqq \,}\frac{3}{2}\lambda ^{-1}$, we have

$$\begin{aligned} |\bar{\mathbf{w }}(r) - \lambda \mathbf{e }_1|&{\,\leqq \,}|\bar{\mathbf{w }}(r) - \bar{\mathbf{w }}(R_1)| + |\bar{\mathbf{w }}(R_1) - \lambda \mathbf{e }_1| \nonumber \\&{\,\leqq \,}C D_\lambda ^{\frac{1}{2}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{1}{2}} + C D_\lambda ^{\frac{1}{2}} {\,\leqq \,}2C D_\lambda ^{\frac{1}{2}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{1}{2}}. \end{aligned}$$

(B.3)

Here we have used that $\log \frac{R_1}{r} {\,\leqq \,}\log \frac{4}{\lambda r} {\,\leqq \,}C \log \frac{2}{\lambda r}$ and $\log \frac{2}{\lambda r} {\,\geqq \,}c$ for any $r {\,\leqq \,}\frac{3}{2}\lambda ^{-1}$, for some absolute positive constants c, C. Combining (B.2) and (B.3), we obtain

$$\begin{aligned} \int _0^{2\pi } |\mathbf{w } - \lambda \mathbf{e }_1|^2 \text {d}\theta&{\,\leqq \,}2\int _0^{2\pi } |\mathbf{w } - \bar{\mathbf{w }}|^2 \text {d}\theta + 2\int _0^{2\pi } |\bar{\mathbf{w }} - \lambda \mathbf{e }_1|^2 \text {d}\theta \\&{\,\leqq \,}2D_\lambda + C D_\lambda \log \frac{2}{\lambda r}\\&{\,\leqq \,}C D_\lambda \log \frac{2}{\lambda r} \end{aligned}$$

for $1{\,\leqq \,}r {\,\leqq \,}\frac{3}{2}\lambda ^{-1}$. Integrating the above in r with respect to the measure rdr gives

$$\begin{aligned} \int _{\Omega _{\frac{1}{2}r, \frac{3}{2}r}} |\mathbf{w } - \lambda \mathbf{e }_1|^2 \, \text {d}x\text {d}y {\,\leqq \,}C r^2 D_\lambda \log \frac{2}{\lambda r} \end{aligned}$$

for any $2{\,\leqq \,}r {\,\leqq \,}\lambda ^{-1}$. (Recall our notation $\Omega _{r_1, r_2} := \{z: r_1<|z|<r_2\}$.) Now we use Ladyzhenskaya’s inequality to obtain

$$\begin{aligned} \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\mathbf{w }-\lambda \mathbf{e }_1|^4 \, \text {d}x\text {d}y&{\,\leqq \,}C\, \int _{\Omega _{\frac{1}{2}r, \frac{3}{2}r}} |\mathbf{w }-\lambda \mathbf{e }_1|^2 \, \text {d}x\text {d}y \ \Big ( \int _{\Omega _{\frac{1}{2}r, \frac{3}{2}r}} |\nabla \mathbf{w }|^2 \, \text {d}x\text {d}y \nonumber \\&\quad \quad \quad \quad \quad +\, \frac{1}{r^2} \int _{\Omega _{\frac{1}{2}r, \frac{3}{2}r}} |\mathbf{w }-\lambda \mathbf{e }_1|^2 \, \text {d}x\text {d}y \Big ) \nonumber \\&{\,\leqq \,}C r^2 D_\lambda ^2 \left( \log \frac{2}{\lambda r}\right) ^2 \end{aligned}$$

(B.4)

for any $2{\,\leqq \,}r {\,\leqq \,}\lambda ^{-1}$. By Hölder’s inequality and (B.4), we have

$$\begin{aligned} \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\mathbf{w } \cdot \nabla \mathbf{w }|^{\frac{4}{3}} \, \text {d}x\text {d}y&{\,\leqq \,}\left( \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\mathbf{w }|^4 \, \text {d}x\text {d}y \right) ^{\frac{1}{3}} \left( \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\nabla \mathbf{w }|^2 \, \text {d}x\text {d}y \right) ^{\frac{2}{3}} \nonumber \\&\lesssim \left( \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\mathbf{w }-\lambda \mathbf{e }_1|^4 \, \text {d}x\text {d}y \right) ^{\frac{1}{3}} \left( \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\nabla \mathbf{w }|^2 \, \text {d}x\text {d}y \right) ^{\frac{2}{3}} \nonumber \\&\quad +\, \left( \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} \lambda ^4 \, \text {d}x\text {d}y \right) ^{\frac{1}{3}} \left( \int _{\Omega _{\frac{2}{3}r, \frac{4}{3}r}} |\nabla \mathbf{w }|^2 \, \text {d}x\text {d}y \right) ^{\frac{2}{3}} \nonumber \\&{\,\leqq \,}C r^{\frac{2}{3}} D_\lambda ^{\frac{4}{3}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{2}{3}} + C r^{\frac{2}{3}} \lambda ^{\frac{4}{3}} D_\lambda ^{\frac{2}{3}} \nonumber \\&{\,\leqq \,}C r^{\frac{2}{3}} \lambda ^{\frac{4}{3}} D_\lambda ^{\frac{2}{3}} \max \left\{ |\log \lambda |^{-\frac{2}{3}} \left( \log \frac{2}{\lambda r} \right) ^{\frac{2}{3}}, 1 \right\} \nonumber \\&{\,\leqq \,}C r^{\frac{2}{3}} \lambda ^{\frac{4}{3}} D_\lambda ^{\frac{2}{3}} \end{aligned}$$

(B.5)

for any $2{\,\leqq \,}r {\,\leqq \,}\lambda ^{-1}$. Local regularity theory for Stokes system as shown in Section 2.3 yields the estimate

$$\begin{aligned} \Vert \nabla ^2 \mathbf{w }\Vert _{L^\frac{4}{3}(\Omega _{\frac{3}{4}r, \frac{5}{4}r})}&{\,\leqq \,}C \Big ( \frac{1}{r^2}\Vert \mathbf{w } - \lambda \mathbf{e }_1\Vert _{L^\frac{4}{3}(\Omega _{\frac{2}{3}r, \frac{4}{3}r})} + \frac{1}{r} \Vert \nabla \mathbf{w }\Vert _{L^\frac{4}{3}(\Omega _{\frac{2}{3}r, \frac{4}{3}r})} \\&\quad +\, \Vert \mathbf{w }\cdot \nabla \mathbf{w }\Vert _{L^\frac{4}{3}(\Omega _{\frac{2}{3}r, \frac{4}{3}r})} \Big ) \end{aligned}$$

where C is independent of r. Applying (B.4), (3.1) and (B.5) to the above inequality, we obtain

$$\begin{aligned} \Vert \nabla ^2 \mathbf{w }\Vert _{L^\frac{4}{3}(\Omega _{\frac{3}{4}r, \frac{5}{4}r})} {\,\leqq \,}C r^{-\frac{1}{2}} D_\lambda ^{\frac{1}{2}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{1}{2}} \end{aligned}$$

for any $2 {\,\leqq \,}r{\,\leqq \,}\lambda ^{-1}$, which clearly implies

$$\begin{aligned} \Vert \nabla ^2 \mathbf{w }\Vert _{L^1(\Omega _{\frac{3}{4}r, \frac{5}{4}r})}{\,\leqq \,}C D_\lambda ^{\frac{1}{2}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{1}{2}}. \end{aligned}$$

Now, Sobolev space theory (see, e.g., [2, Lemma 4.3] ) gives the following bound for the variation of $\mathbf{w }$:

$$\begin{aligned} \text {diam}\, \mathbf{w }(\Omega _{\frac{3}{4}r, \frac{5}{4}r})&{\,\leqq \,}C \Big ( \Vert \nabla \mathbf{w }\Vert _{L^2(\Omega _{\frac{3}{4}r, \frac{5}{4}r})} + \Vert \nabla ^2 \mathbf{w }\Vert _{L^1(\Omega _{\frac{3}{4}r, \frac{5}{4}r})} \Big ) \nonumber \\&{\,\leqq \,}C D_\lambda ^{\frac{1}{2}} \left( \log \frac{2}{\lambda r}\right) ^{\frac{1}{2}}, \end{aligned}$$

(B.6)

for any $2 {\,\leqq \,}r{\,\leqq \,}\lambda ^{-1}$. Together with (B.3), (B.6) gives the desired bound

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

(B.7)

in the region $2 {\,\leqq \,}r{\,\leqq \,}\lambda ^{-1}$. To finish the proof, we point out that for the region $\{r {\,\leqq \,}2\} \cap {\mathcal {E}}$, due to Lemma 11, (B.7) also holds true. $\square$