Balanced-Viscosity solutions for multi-rate systems

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Date
2014
Volume
2001
Issue
Journal
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Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients α and , where 0<<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.

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Keywords
Generalized gradient systems, vanishing-viscosity approach, energy-dissipation principle, jump curves
Citation
Mielke, A., Rossi, R., & Savaré, G. (2014). Balanced-Viscosity solutions for multi-rate systems (Vol. 2001). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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