Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

Fang Hua Lin, Jiajun Tong

Research output: Contribution to journalArticlepeer-review

Abstract

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear nonlocal equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an H5/2-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parametrized circular configuration, then a global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as t → + ∞. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions.

Original languageEnglish (US)
Pages (from-to)159-226
Number of pages68
JournalCommunications on Pure and Applied Mathematics
Volume72
Issue number1
DOIs
StatePublished - Jan 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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