Second order approximations of  quasistatic evolution problems  in finite dimension

Virginia Agostiniani

doi:10.3934/dcds.2012.32.1125

Article Contents

2012, Volume 32, Issue 4: 1125-1167. Doi: 10.3934/dcds.2012.32.1125

This issue Previous Article A convex-concave elliptic problem with a parameter on the boundary condition Next Article Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control

Second order approximations of quasistatic evolution problems in finite dimension

Virginia Agostiniani^1,

1.
via Bonomea 265, 34136 Trieste

Received: October 31, 2010

Revised: March 31, 2011

Published: March 2012

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Abstract

In this paper, we study the limit, as $\epsilon$ goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^{\epsilon}(t)+\epsilon B\dot u^{\epsilon}(t)+\nabla_xf(t,u^{\epsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\epsilon}(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

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Mathematics Subject Classification: Primary: 74H10, 34K18, 34C37; Secondary: 34K26.

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