N.T., Hong Kong '
author:
- Daomin Cao
- Zhongyuan Liu
- Juncheng Wei
title:  Regularization of point vortices for the Euler equation in dimension two 
---

Introduction and main results
=============================

The incompressible Euler equations $$\label{1.2}
\begin{cases}
 \mathbf{v}_t+(\mathbf{v}\cdot\nabla)\mathbf{v}=-\nabla P,\\
\nabla\cdot\mathbf{v}=0,
\end{cases}$$ describe the evolution of the velocity $\mathbf{v}$ and the pressure $P$ in an incompressible flow. In $\mathbb{R}^2$, the vorticity of the flow is defined by $\omega=\nabla\times\mathbf{v}:=\partial_1
v_2-\partial_2 v_1$, which satisfies the equation $$\omega_t+\mathbf{v}\cdot\nabla\omega=0.$$

Suppose that $\omega$ is known, then the velocity $\mathbf{v}$ can be recovered by Biot-Savart law as following: $$\mathbf{v}=\omega\,*\frac{1}{2\pi}\frac{-x^\bot}{|x|^2},$$ where $x^\bot=(x_2,\,-x_1)$ if $x=(x_1,\,x_2)$. One special singular solutions of Euler equations is given by $\omega=\sum^m_{i=1}\kappa_i\delta_{x_i(t)}$, which is related $$\mathbf{v}=-\sum^m_{i=1}\frac{\kappa_i}{2\pi}\frac{(x-x_i(t))^\bot}{|x-x_i(t)|^2}.$$ and the positions of the vortices $x_i: \mathbb{R}\rightarrow
\mathbb{R}^2$ satisfy the following Kirchhoff law: $$\kappa_i\,\frac{dx_i}{dt}=(\nabla_{x_i}\mathcal{W})^\bot$$ where $\mathcal{W}$ is the so called Kirchhoff-Routh function defined by $$\mathcal{W}(x_1,\cdots, x_m)=\frac{1}{2}\sum_{i\neq
j}^m\frac{\kappa_i\kappa_j}{2\pi}\log\frac{1}{|x_i-x_j|}.$$

In simply-connected bounded domain $\Omega\subset \mathbb{R}^2$, similar singular solutions also exist. Suppose that the normal component of $\mathbf{v}$ vanishes on $\partial\Omega$, then the Kirchhoff-Routh function is

$$\label{W1}
\mathcal{W}(x_1,\cdots, x_m)=\frac{1}{2}\sum_{i\neq
j}^m{\kappa_i\kappa_j}G(x_i,\,x_j)+
\frac{1}{2}\sum_{i=1}^m{\kappa_i^2}H(x_i,\,x_i),$$

where $G$ is the Green function of $-\Delta$ on $\Omega$ with 0 Dirichlet boundary condition and $H$ is its regular part (the Robin function).

Let $v_n$ be the outward 