2D Peskin Problems of an Immersed Elastic Filament in Stokes Flow

Abstract

In the work that follows we investigate a class of problems where a one dimensional closed elastic structure is immersed in a plane of steady Stokes flow. The dynamics are governed by a boundary integral equation describing the configuration of the immersed structure. Depending on the choice of elasticity law, we break our class into either a semilinear or fully nonlinear system of equations. In the nonlinear setting, we prove that the linearization of the system generates an analytic semigroup and use it to prove local existence and uniqueness in low regularity H\"{o}lder spaces. In the semilinear setting, we remove the principle operator via small scale decomposition and use it to build similar local existence results. Further, we establish spatial smoothness of solutions by careful estimates on a class of commutators. Using these regularity results, we are able to establish that the only equilibria of the system are uniformly parameterized circles which we then prove nonlinear stability about. Finally, we identify a quantity which classifies global-in-time behavior.