具有弯曲和拉伸能量的Stokes浸入边值问题的稳定性
Stability of the Stokes Immersed Boundary Problem with Bending and Stretching Energy
摘要：
研究了二维Stokes流中具有弯曲能和拉伸能的一维闭合弹性弦的运动。本文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。借助于斯托克斯方程的基本解，我们将问题转化为一个抛物型方程组，称之为轮廓动力系统。在初始构型非自相交和良好伸展的假设下，我们建立了索伯列夫空间中自由边界问题的局部适定性。当初始构型足够接近平衡态（即均匀参数化圆）时，我们证明了解可以全局扩展，并且全局解将以$\to$ + $\infty$的形式指数收敛到平衡态.
英文摘要：
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 t $\to$ +$\infty$.
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