Well-Posedness of the 3D Peskin Problem

TL;DR

This paper establishes the well-posedness and instant smoothing of a 3D elastic membrane immersed in steady Stokes flow, modeled by a free boundary problem with nonlinear elastic laws, using boundary integral methods.

Contribution

It introduces the 3D Peskin problem, derives its equations, and proves well-posedness and regularity results for the first time.

Findings

The problem admits a boundary integral reduction.

The elastic membrane becomes instantly smooth in time.

Well-posedness holds in low-regularity H"older spaces.

Abstract

This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H\"older spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.