  In this paper we study a toy model of the Peskin problem that captures the
motion of the full Peskin problem in the normal direction and discards the
tangential elastic stretching contributions. This model takes the form of a
fully nonlinear scalar contour equation. The Peskin problem 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. 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.
