  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 $\to$ +$\infty$.
