Global Solutions to the Tangential Peskin Problem in 2-D

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 supercritical 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.

Well-Posedness of the 3D Peskin Problem

“…The work [23] considers a model for the normal component and shows the existence of global solutions for Lipschitz data near the equilibrium. Very recently, [52] derives a PDE to model the tangential effects of the Peskin problem in the case of an infinitely long and straight string and obtains global solutions with initial data in the energy class. Moreover, the author presents many connections of the model with well-known one-dimensional PDEs.…”

“…Since the recent breakthrough works [34] and [37], which provided the strong solution theory for the problem of an immersed elastic string in a two-dimensional fluid, the so-called 2D Peskin problem has attracted a lot of attention [8,9,23,25,33,51,52]. In this paper, we initiate the study of its three-dimensional counterpart.…”