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

Abstract

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
Volume8
Issue number3
DOIs
StatePublished - Jan 1 2010

Fingerprint

Local Existence
Geometric Nonlinearity
Existence and Uniqueness
Membrane
Nonlinearity
Membranes
Fluid
Fluids
Blowing-up
Incompressibility
Wave equations
Fluid Dynamics
Blow molding
Fluid dynamics
Elliptic Equations
Coupled System
Compactness
Elasticity
Wave equation
Vanish

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space. / Hu, Dan; Song, Peng; Zhang, Pingwen.

In: Communications in Mathematical Sciences, Vol. 8, No. 3, 01.01.2010, p. 783-796.

Research output: Contribution to journalArticle

@article{3ee7f83580b74a8ca7cfdf494f247994,
title = "Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space",
abstract = "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.",
keywords = "And bending elasticity, Existence, Incompressible, Membrane, Uniqueness",
author = "Dan Hu and Peng Song and Pingwen Zhang",
year = "2010",
month = "1",
day = "1",
doi = "10.4310/CMS.2010.v8.n3.a9",
language = "English (US)",
volume = "8",
pages = "783--796",
journal = "Communications in Mathematical Sciences",
issn = "1539-6746",
publisher = "International Press of Boston, Inc.",
number = "3",

}

TY - JOUR

T1 - Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space

AU - Hu, Dan

AU - Song, Peng

AU - Zhang, Pingwen

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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.

AB - 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.

KW - And bending elasticity

KW - Existence

KW - Incompressible

KW - Membrane

KW - Uniqueness

UR - http://www.scopus.com/inward/record.url?scp=77954642416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954642416&partnerID=8YFLogxK

U2 - 10.4310/CMS.2010.v8.n3.a9

DO - 10.4310/CMS.2010.v8.n3.a9

M3 - Article

AN - SCOPUS:77954642416

VL - 8

SP - 783

EP - 796

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 3

ER -