Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

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.