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September 2010 Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space
Dan Hu, Peng Song, Pingwen Zhang
Commun. Math. Sci. 8(3): 783-796 (September 2010).

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.

Citation

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Dan Hu. Peng Song. Pingwen Zhang. "Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space." Commun. Math. Sci. 8 (3) 783 - 796, September 2010.

Information

Published: September 2010
First available in Project Euclid: 25 August 2010

zbMATH: 1206.35009
MathSciNet: MR2730331

Subjects:
Primary: 35M13 , 65M12 , 92C17

Keywords: bending elasticity , existence , incompressible , Membrane , uniqueness

Rights: Copyright © 2010 International Press of Boston

Vol.8 • No. 3 • September 2010
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