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

Abstract

In this paper we study the Peskin problem in 2D, which describes the dynamics of a 1D closed elastic structure immersed in a steady Stokes flow. We prove the local well-posedness for arbitrary initial configuration in V M O 1 satisfying the well-stretched condition, and the global well-posedness when the initial configuration is sufficiently close to an equilibrium in BM O 1 . The global-in-time solution will converge to an equilibrium exponentially as t → +∞. This is the first well-posedness result for the Peskin problem with non-Lipschitz initial data.

Long time interface dynamics for gravity Stokes flow

“…One of the classical methods to deal with free boundary problems is to exploit potential theory in order to reformulate the problem into a new contour dynamics equation, which will be typically nonlocal and strongly non-linear. Let us mention that this kind of approach has been extensively and successfully used in other free boundary problems in fluid dynamics to show well-posedness (see [3,8] for the vortex patch, [11] for water waves, [30,18,17] for the SQG sharp-front, [12,16] for the Muskat problem and [19,9] for the Peskin problem). Furthermore, it has been applied to prove singularity formation for the water waves, the SQG sharp-front and the Muskat problem [6,4,29,18,5].…”

“…Furthermore, both problems can be studied as a nonlinear and nonlocal partial differential equation using a contour dynamics approach similar to the one that we use in this paper. On the one hand, the Peskin problem models the dynamics of an elastic filament immersed in a Stokes fluid [43,19,34,9]. However, in the problem under consideration in this paper, the gravity driven Stokes problem, the dynamics is induced by the gravity force.…”

Well-Posedness of the 3D Peskin Problem

“…The result [8] shows the local well-posedness and smoothing for general data in the critical Besov space B 3 2 2,1 , including the case of nonlinear elastic law. The sharpest result in terms of regularity appeared in [9], where the semilinear 2D Peskin problem is shown to be well-posed in B 1 8,8 , and thus with possibly non-Lipschitz curves.…”

“…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.…”

Critical local well-posedness for the fully nonlinear Peskin problem

“…Recently, Gancedo, Belinchón and Scrobogna [20] studied a toy model of the Peskin problem and proved global existence and uniqueness in the critical Lipschitz space. Then more recently, Chen and Nguyen were able to prove local well-posedness for (1.7) whenever X 1 0 is in VMO using estimates on the fundamental solution of p´∆q 1 2 and interpolation results, and they further prove global existence when X 1 0 is in BMO for initial data that is close to equilibrium [11].…”