A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

Alexander Mielke; Michael Ortiz

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 14, Issue: 3, page 494-516
  • ISSN: 1292-8119

Abstract

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This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as a weighted dissipation-energy functional with a weight decaying with a rate 1 / ϵ . The corresponding Euler-Lagrange equation leads to an elliptic regularization of the original evolutionary problem. The Γ-limit of these functionals for ϵ → 0 is highly degenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.

How to cite

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Mielke, Alexander, and Ortiz, Michael. "A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 494-516. <http://eudml.org/doc/90880>.

@article{Mielke2007,
abstract = { This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as a weighted dissipation-energy functional with a weight decaying with a rate $1/\epsilon$. The corresponding Euler-Lagrange equation leads to an elliptic regularization of the original evolutionary problem. The Γ-limit of these functionals for $\epsilon\to 0$ is highly degenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation. },
author = {Mielke, Alexander, Ortiz, Michael},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Weighted energy-dissipation functional; incremental minimization problems; relaxation of evolutionary problems; rate-independent processes; energetic solutions; weighted energy-dissipation functional},
language = {eng},
month = {12},
number = {3},
pages = {494-516},
publisher = {EDP Sciences},
title = {A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems},
url = {http://eudml.org/doc/90880},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Mielke, Alexander
AU - Ortiz, Michael
TI - A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/12//
PB - EDP Sciences
VL - 14
IS - 3
SP - 494
EP - 516
AB - This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as a weighted dissipation-energy functional with a weight decaying with a rate $1/\epsilon$. The corresponding Euler-Lagrange equation leads to an elliptic regularization of the original evolutionary problem. The Γ-limit of these functionals for $\epsilon\to 0$ is highly degenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.
LA - eng
KW - Weighted energy-dissipation functional; incremental minimization problems; relaxation of evolutionary problems; rate-independent processes; energetic solutions; weighted energy-dissipation functional
UR - http://eudml.org/doc/90880
ER -

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