The Peskin problem with $BMO^1$ initial data

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 $VMO^1$ satisfying the well-stretched condition, and the global well-posedness when the initial configuration is sufficiently close to an equilibrium in $BMO^1$. The global-in-time solution will converge to an equilibrium exponentially as $trightarrow+infty$. This is the first well-posedness result for the Peskin problem with non-Lipschitz initial data.