Global existence in the Lipschitz class for the N-Peskin problem

Abstract

In this paper we study the Peskin problem. This is a fluid-structure interaction problem that describes the motion of an elastic rod immersed in an incompressible Stokes fluid. We prove global in time existence of solution for initial data in the critical Lipschitz space. To obtain this result we use a new contour dynamic formulation which reduces the system to a scalar equation. Using a new decomposition together with cancellation properties, pointwise methods allow us to obtain the desired estimates in the Lipschitz class. Moreover, we perform energy estimates in order to obtain that the solution lies in the space L 2 [0, T ] ; H 3/2 to satisfy the contour equation pointwise.

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

“…Their result holds for two fluids with different viscosity. More recently, Gancedo, Belinchón and Scrobogna [19] considered a toy model of the Peskin problem and proved a global existence result in the critical Lipschitz space.…”

Global Solutions to the Tangential Peskin Problem in 2-D

“…(2.11) gives the exact string dynamics in the special setting considered here, as no approximation is made in the derivation. This is in contrast to the scalar model in [47], where the tangential deformation of the string should have made a difference to the evolution but gets ignored purposefully.…”

“…García-Juárez, Mori, and Strain considered the case where the fluids in the interior and the exterior of the string can have different viscosities, and they showed global well-posedness for initial data of medium size in the critical Wiener space F 1,1 (T) [46]. In [47], Gancedo, Granero-Belinchón, and Scrobogna introduced a toy scalar model that captures the string motion in the normal direction only, for which they proved global well-posedness for small initial data in the critical Lipschitz class W 1,∞ (T). Recently, Chen and Nguyen established well-posedness of the original 2-D Peskin problem in the critical space Ḃ1…”

Well-posedness and applications of classical elastohydrodynamics for a swimming filament

“…As a byproduct, we obtain nonlinear stability of the straight filament. The filament evolution in this setting may be regarded as a simpler version of the Peskin problem (immersed boundary method), which has been the subject of many recent PDE works [13,14,29,40,54]. This work also follows a recent program by the authors to place nonlocal slender body theory on firm theoretical footing [38,39,36,37,44].…”