  In this paper we study the Peskin problem in 2D, which describes the dynamics
of a 1D closed elastic structure immersed in a steady Stokes flow. We prove the
local well-posedness for arbitrary initial configuration in $(C^2)^{\dot
B^1_{\infty,\infty}}$ satisfying the well-stretched condition, and the global
well-posedness when the initial configuration is sufficiently close to an
equilibrium in $\dot B^1_{\infty,\infty}$. Here $(C^2)^{\dot
B^1_{\infty,\infty}}$ is the closure of $C^2$ in the Besov space $\dot
B^1_{\infty,\infty}$. The global-in-time solution will converge to an
equilibrium exponentially as $t\rightarrow+\infty$. This is the first
well-posedness result for the Peskin problem with non-Lipschitz initial data.
