Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space

Dan Hu, Peng Song, Pingwen Zhang

Research output: Contribution to journalArticle


The dynamics of a membrane is a coupled system of a moving elastic surface and an incompressible membrane fluid. The difficulties in analyzing such a system include the nonlinearity of the curved space (geometric nonlinearity), the nonlinearity of the fluid dynamics (fluid nonlinearity), and the coupling to the surface incompressibility. In the two-dimensional case, the fluid vanishes and the system reduces to a coupling of a wave equation and an elliptic equation. Here we prove the local existence and uniqueness of the solution to the system by constructing a suitable discrete scheme and proving the compactness of the discrete solutions. The risk of blowing up due to the geometric nonlinearity is overcome by the bending elasticity.

Original languageEnglish (US)
Pages (from-to)783-796
Number of pages14
JournalCommunications in Mathematical Sciences
Issue number3
StatePublished - Sep 2010



  • And bending elasticity
  • Existence
  • Incompressible
  • Membrane
  • Uniqueness

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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