I have a question regarding the uniqueness of the potential flow past a cylinder.

Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \mathbb{R}^2 : \vert x \vert < R \}$ for some radius $R > 0$. The velocity potential $\varphi(x)$ is then subject to the boundary value problem

\begin{equation} \begin{aligned} \Delta_x \, \varphi(x) &= 0 \qquad &&\text{in} \quad \mathbb{R}^2 \setminus \overline{B_R} \; , \\ \frac{x}{\vert x \vert} \cdot \nabla_x \, \varphi(x) &= 0 \qquad &&\text{on} \quad \partial B_R \; , \\ \end{aligned} \end{equation}

together with the limiting behaviour $\nabla_x \, \varphi(x) \to (v_\infty, 0)$ as $\vert x \vert \to \infty$ for some $v_\infty > 0$. In most hydrodynamic lectures the general solution to this problem is given as the superposition of a uniform flow and doublet in $x_1$-direction,

\begin{equation} \varphi(x) = v_\infty \left( x_1 + \frac{R^2 x_1}{x_1^2 + x_2^2} \right) \; . \end{equation}

While I see that the given solution solves the stated problem I do not see how to proof uniqueness or wether this is even possible.

As far as I know the exterior Neumann problem of the Laplace equation for some bounded (and for simplicity connected) domain $\Omega \subset \mathbb{R}^2$ with outer normal $\nu$,

\begin{equation} \begin{aligned} \Delta u&= 0 \qquad &&\text{in} \quad \mathbb{R}^2 \setminus \overline{\Omega} \; , \\ \frac{\partial u}{\partial \nu} &= 0 \qquad &&\text{on} \quad \partial \Omega \; , \\ \end{aligned} \end{equation}

together with the limiting behaviour $\vert u(x) \vert / \ln(\vert x \vert) \to 0$ as $\vert x \vert \to \infty$ has the unique harmonic solution $u = const.$ on $\mathbb{R}^2 \setminus \Omega$ up to an additive constant.

Since for the above solution $\vert \varphi(x) \vert / \ln(\vert x \vert)$ diverges as $\vert x \vert \to \infty$ I wonder wether it is the unique harmonic solution and how to proof/contradict it.


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