Critical local well-posedness for the fully nonlinear Peskin problem

TL;DR

This paper establishes local well-posedness and smoothing effects for the fully nonlinear Peskin problem, modeling an elastic string in a viscous fluid, using a novel boundary integral formulation and cancellation techniques.

Contribution

It proves local well-posedness in critical Besov spaces for the fully nonlinear Peskin problem and introduces a new boundary integral formulation with cancellation structures.

Findings

Proved local well-posedness in $ ext{dot}B^{3/2}_{2,1}$ space.

Established optimal higher order smoothing effects.

Developed a new formulation and cancellation structure for the boundary integral equation.

Abstract

We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\dot{B}^{3/2}_{2,1}(\mathbb{T} ; \mathbb{R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.