Well-posedness of Hydrodynamics on the Moving Elastic Surface

TL;DR

This paper proves the local existence and uniqueness of solutions for a coupled model of a moving elastic surface and an incompressible fluid within a curved geometry, using isothermal coordinates reformulation.

Contribution

It introduces a new approach to establish well-posedness for the hydrodynamics on a moving elastic surface by reformulating the model in isothermal coordinates.

Findings

Established local existence and uniqueness of solutions.

Reformulated the model in isothermal coordinates for analysis.

Addressed the construction of an iterative scheme for the coupled system.

Abstract

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.