Global existence in the Lipschitz class for the N-Peskin problem

In this paper we study the Peskin problem. This is a fluid-structure interaction problem that describes the motion of an elastic rod immersed in an incompressible Stokes fluid. We prove global in time existence of solution for initial data in the critical Lipschitz space. To obtain this result we use a new contour dynamic formulation which reduces the system to a scalar equation. Using a new decomposition together with cancellation properties, pointwise methods allow us to obtain the desired estimates in the Lipschitz class. Moreover, we perform energy estimates in order to obtain that the solution lies in the space $L^2 \left( [0,T];H^{3/2} \right) $ to satisfy the contour equation pointwise.