Well‐Posedness and Global Behavior of the Peskin Problem of an Immersed Elastic Filament in Stokes Flow

Abstract

We consider the problem of a one-dimensional elastic filament immersed in a two-dimensional steady Stokes fluid. Immersed boundary problems in which a thin elastic structure interacts with a surrounding fluid are prevalent in science and engineering, a class of problems for which Peskin has made pioneering contributions. Using boundary integrals, we first reduce the fluid equations to an evolution equation solely for the immersed filament configuration. We then establish local well-posedness for this equation with initial data in low-regularity Hölder spaces. This is accomplished by first extracting the principal linear evolution by a small-scale decomposition and then establishing precise smoothing estimates on the nonlinear remainder. Higher regularity of these solutions is established via commutator estimates with error terms generated by an explicit class of integral kernels. Furthermore, we show that the set of equilibria consists of uniformly parametrized circles and prove nonlinear stability of these equilibria with explicit exponential decay estimates, the optimality of which we verify numerically. Finally, we identify a quantity that respects the symmetries of the problem and controls global-in-time behavior of the system.

Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

“…Although the focus of the present work is mostly on the analysis and approximation of the proposed approach, we stress that it aims to build the mathematical foundations for tackling various applications involving 3D-1D mixed-dimensional PDEs, such as fluid-structure interaction of slender bodies [26], microcirculation and lymphatics [29,33], subsurface flow models with wells [8], and the electrical activity of neurons.…”

Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well‐Posedness, Singular Limit, and Error Estimates

“…Regularity of u recovered from X.s; t/ is studied in [39]. In a parallel work, Mori, Rodenberg, and Spirn [23] establish similar local and global well-posedness results for (1.5) in C 1;spaces. They also show improved regularity of X.s; t/ for positive time and a blowup criterion.…”

“…Recall that in the analysis of the original problem with the Hookean elasticity [15,23], (1.1)-(1.4) is first reduced (under some assumptions) to the contour dynamic equation (1.5), and it is then rewritten as @ t X h LX g g X . Here LX h 1 4 .…”

“…The first term in the last line of (5.8) provides damping over all frequencies, which is the very fact used in [15,23] to establish well-posedness of the original unregularized problem. The other two terms can be bounded by applying commutator estimates (see, e.g., [13]).…”

A posteriori error analysis of fluid–structure interactions: Time dependent error

“…Discussions of the well-posedness of Stokes equations and numerical convergence results for the MRS can be found in [7,8,14,11,18]. However, it appears that the well-posedness of the quasi-steady state MRS for general forces has not been studied (though two recent studies on specific examples in two dimensions exist [17,15]). On the other hand, for the spatially discrete systems considered in this article, well-posedness is guaranteed by the Picard existence and uniqueness theorem (for short times) as long as the force operator is a Lipschitz continuous function of x(α, t) and t [13, §3].…”