Title: Solvability of the Stokes Immersed Boundary Problem in Two Dimensions,
Abstract: We study coupled motion of a 1-D closed elastic string immersed in a 2-D
Stokes flow, known as the Stokes immersed boundary problem in two dimensions.
Using the fundamental solution of the Stokes equation and the Lagrangian
coordinate of the string, we write the problem into a contour dynamic
formulation, which is a nonlinear non-local equation solely keeping track of
evolution of the string configuration. We prove existence and uniqueness of
local-in-time solution starting from an arbitrary initial configuration that is
an $H^{5/2}$-function in the Lagrangian coordinate satisfying the so-called
well-stretched assumption. We also prove that when the initial string
configuration is sufficiently close to an equilibrium, which is an evenly
parameterized circular configuration, then global-in-time solution uniquely
exists and it will converge to an equilibrium configuration exponentially as
$t\rightarrow +\infty$. The technique in this paper may also apply to the
Stokes immersed boundary problem in three dimensions.