Well-posedness of two-dimensional hydroelastic waves

Abstract

A well-posedness theory for the initial-value problem for hydroelastic waves in two spatial dimensions is presented. This problem, which arises in numerous applications, describes the evolution of a thin elastic membrane in a two-dimensional (2D) potential flow. We use a model for the elastic sheet that accounts for bending stresses and membrane tension, but which neglects the mass of the membrane. The analysis is based on a vortex sheet formulation and, following earlier analyses and numerical computations in 2D interfacial flow with surface tension, we use an angle–arclength representation of the problem. We prove short-time well-posedness in Sobolev spaces. The proof is based on energy estimates, and the main challenge is to find a definition of the energy and estimates on high-order non-local terms so that an a priori bound can be obtained.

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

“…We refer the reader to [35,36,39] where an extensive list of references on these problems can be found. We also point to several recent results on problem with structures with more complicated energies interacting with the surrounding fluid [2,6,8,24,29,32,33].…”

“…The analytical study of the one-phase problem, in which the fluid equations (Stokes or Navier-Stokes) are satisfied in the interior region i only, was initiated by Solonnikov [41] and has since been taken up by many authors. We also point to several recent results on problem with structures with more complicated energies interacting with the surrounding fluid [2,6,8,24,29,32,33]. We refer the reader to [35,36,39] where an extensive list of references on these problems can be found.…”

Periodic traveling interfacial hydroelastic waves with or without mass

“…The evolution of the interface is also determined by the behavior of the vortex sheet-strength γ (α, t), which can be written in terms of the jump in tangential velocity across the surface. Using a model which combines those used in [8] and [16], we assume the jump in pressure across the interface to be…”

“…This system is more suitable for large surface deformations than simpler models such as linear or Kirchoff-Love models. The second author, Siegel, and Liu have shown that the initial value problems for these Cosserat-type hydroelastic waves are well-posed in Sobolev spaces [8], [16]. Toland and Baldi and Toland have proved existence of periodic traveling hydroelastic water waves with and without mass including studying secondary bifurcations [25], [26], [10], [11].…”

The Viscous Surface Wave Problem with Generalized Surface Energies

“…For two-dimensional irrotational two-fluid flows, Liu-Ambrose [LA17] proved wellposedness and Akers-Ambrose-Sulon [AAS17] constructed traveling wave solutions for a two-fluid model. The one-fluid model was studied by Ambrose-Siegel [AS17].…”