component of the velocity $\mathbf{v}$ on the boundary $\partial\Omega$, then we see that $\int_{\partial\Omega}v_n=0$ due to the fact that $\nabla\cdot\mathbf{v}=0$. Suppose that $\mathbf{v}_0$ is the unique harmonic field whose normal component on the boundary $\partial\Omega$ is $v_n$, then $\mathbf{v}_0$ satisfies

$$\label{v0}
\begin{cases}
 \nabla\cdot\mathbf{v}_0=0,\,\,\text{in}\,\Omega,\\
\nabla\times\mathbf{v}_0=0,\,\,\text{in}\,\Omega,\\
n\cdot\mathbf{v}_0=v_n,\,\, \text{on}\,\partial\Omega.
\end{cases}$$

If $\Omega$ is simply-connected, then $\mathbf{v}_0$ can be written $\mathbf{v}_0=(\nabla\psi_0)^\bot$, where the stream function $\psi_0$ is determined up to a constant by $$\label{psi}
\begin{cases}
 -\Delta \psi_0=0,\,\,\text{in}\,\Omega,\\
-\displaystyle\frac{\partial\psi_0}{\partial\tau}=v_n,\,\,
\text{on}\,\partial\Omega,
\end{cases}$$ where $\frac{\partial\psi_0}{\partial\tau}$ denotes the tangential derivative on $\partial\Omega$. The Kirchhoff-Routh function associated to the vortex dynamics becomes(see Lin [@Lin])

$$\label{KR}
\mathcal{W}(x_1,\cdots,x_m)=\frac{1}{2}\sum_{i\neq
j}^m\kappa_i\kappa _jG(x_i,x_j)+\frac{1}{2}\sum^{m}_{i=1}\kappa^2
_iH(x_i,x_i)+\sum^{m}_{i=1}\kappa_i\psi_0(x_i).$$

It is known that critical points of the Kirchhoff-Routh function $\mathcal{W}$ give rise to stationary vortex points solutions of the Euler equations. As for the existence of critical points of $\mathcal{W}$ given by , we refer to [@BT].

Roughly speaking, there are two methods to construct stationary solutions of the Euler equation, which are the vorticity method and the stream-function method. The vorticity method was first established by Arnold and Khesin [@AK] and further developed by Burton [@B] and Turkington [@T].

The stream-function method consists in observing that if $\psi$ satisfies $-\Delta \psi=f(\psi)$ for some function $f\in
C^1(\mathbb{R})$, then $\mathbf{v}=(\nabla\psi)^\bot$ 