Global Solutions to the Tangential Peskin Problem in 2-D

TL;DR

This paper introduces a new 2-D scalar drift-diffusion equation derived from a specific elastic string problem in fluid flow, proving global existence, regularity, and long-term behavior of solutions.

Contribution

It presents the first global solutions to a super-critical 2-D Peskin problem using an Eulerian approach, expanding understanding of elastic-fluid interactions.

Findings

Existence of global solutions for initial data in the energy class.

Regularity and long-time behavior of solutions established.

Uniqueness proved under additional assumptions.

Abstract

We introduce the tangential Peskin problem in 2-D, which is a scalar drift-diffusion equation with a nonlocal drift. It is derived with a new Eulerian perspective from a special setting of the 2-D Peskin problem where an infinitely long and straight 1-D elastic string deforms tangentially in the Stokes flow induced by itself in the plane. For initial datum in the energy class satisfying natural weak assumptions, we prove existence of its global solutions. This is considered as a super-critical problem in the existing analysis of the Peskin problem based on Lagrangian formulations. Regularity and long-time behavior of the constructed solution is established. Uniqueness of the solution is proved under additional assumptions.