We study the motion of a 1-D closed elastic string with bending and stretching energy immersed in a 2-D Stokes flow. In this paper we introduce the curve's tangent angle function and the stretching function to describe the deferent deformations of the elastic string. These two functions are defined on the arc-length coordinate and the material coordinate respectively. With the help of the fundamental solution of the Stokes equation, we reformulate the problem into a parabolic system which is called the contour dynamic system. Under the non-self-intersecting and well-stretched assumptions on initial configurations, we establish the local well-posedness of the free boundary problem in Sobolev space. When the initial configurations are sufficiently close to the equilibrium state (i.e. an evenly parametrized circle), we prove that the solutions can be extended globally and the global solutions will converge to the equilibrium state exponentially as $\text{t}\to +\infty$.

我们研究了浸入二维斯托克斯流中的具有弯曲和拉伸能量的一维闭合弹性弦的运动。在本文中，我们引入曲线的切角函数和拉伸函数来描述弹性弦的不同变形。这两个函数分别定义在弧长坐标和材料坐标上。借助斯托克斯方程的基本解，我们将问题重新表述为抛物线系统，称为轮廓动力学系统。在初始配置的非自相交和良好拉伸假设下，我们建立了Sobolev空间中自由边界问题的局部适定性。当初始配置足够接近平衡状态（即均匀参数化的圆）时，$\text{吨}\to +\infty$.