The Peskin problem with ${\dot{B}}_{\infty ,\infty }^1$ initial data

Abstract

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 $(C^2{)}^{{\dot{B}}_{\infty ,\infty }^1}$ satisfying the well-stretched condition, and the global well-posedness when the initial configuration is sufficiently close to an equilibrium in ${\dot{B}}_{\infty ,\infty }^1$. Here $(C^2{)}^{{\dot{B}}_{\infty ,\infty }^1}$ is the closure of $C^2$ in the Besov space ${\dot{B}}_{\infty ,\infty }^1$. The global-in-time solution will converge to an equilibrium exponentially as $t\to +\infty$. This is the first well-posedness result for the Peskin problem with non-Lipschitz initial data.

arXiv commentWe add a proof of ${\tilde{\mathcal{G}}}^1={\dot{B}}_{\infty ,\infty }^1$, so the Peskin problem is well-posed in ${\dot{B}}_{\infty ,\infty }^1$. We change the title of our paper to "The Peskin problem with ${\dot{B}}_{\infty ,\infty }^1$ initial data"