Article Contents

2016, Volume 36, Issue 2: 833-849. Doi: 10.3934/dcds.2016.36.833

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Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS

Jingyu Li^1, and
Chuangchuang Liang^2,

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024
2.
Department of Mathematics, Capital Normal University, Beijing 100037

Received: July 2014

Revised: January 2015

Published: May 2017

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Abstract

We study the asymptotic relation of solutions between the hyperbolic equation and the parabolic one over a one-dimensional bounded interval, both of which model a simple electrostatic micro-electro-mechanical system (MEMS) device. The relation is characterized by a limit as a physical parameter representing the strength of inertial forces tends to zero. We call this limit the viscosity dominated limit. It is shown that in this singular limit the solution of the hyperbolic model converges to that of the parabolic one globally in time. Also the higher order terms including the initial layer corrections, as well as the related error estimates, are derived. Furthermore, it is proved that the convergence is valid for global solutions with large initial data.

Keywords:

Mathematics Subject Classification: Primary: 35B25, 35A01; Secondary: 35L20, 74H10.

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