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Your search keyword '"Minimizing movement"' showing total 3 results
Start Over Descriptor "Minimizing movement" Topic viscosity solution
3 results on '"Minimizing movement"'

1. Viscosity solutions for doubly nonlinear evolution equations.

Author

Courte, Luca and Dondl, Patrick

Subjects
*VISCOSITY solutions, *NONLINEAR evolution equations, *ORDINARY differential equations, *DRY friction, *HYSTERESIS loop, *DIFFERENTIAL inclusions
Abstract

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop. [ABSTRACT FROM AUTHOR]

Published
2023
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2. Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach

Author

Giovanni Scilla, Francesco Solombrino, Scilla, G., and Solombrino, F.

Subjects
singular perturbations, Gradient Flow, Variational methods, rate-independent systems, minimizing movement, Balanced Viscosity solutions, crease energy, Scale (ratio), crease energy, variational methods, General Physics and Astronomy, balanced viscosity solutions, 01 natural sciences, Viscosity, gradient flow, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, Limit (mathematics), 0101 mathematics, Mathematical Physics, minimizing movement, Mathematics, 49Q20, 74H10, 34K26, Applied Mathematics, 010102 general mathematics, Mathematical analysis, singular perturbations, rate-independent systems, Statistical and Nonlinear Physics, 010101 applied mathematics, Jump, Viscosity solution, Balanced flow, Quasistatic process, Analysis of PDEs (math.AP)
Abstract

We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $\tau$. When the ratio $\frac{\varepsilon}{\tau}$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio between the scales, which is of a different kind. In this case, a discrete interfacial energy is optimized at jump times.

Published
2017

3. On the energy of a flow arising in shape optimization

Author

Pierre Cardaliaguet, Olivier Ley, Laboratoire de Mathématiques (LM-Brest), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), The authors are partially supported by the ACI grant JC 1041 ``Mouvements d'interface avec termes non-locaux' from the French Ministry, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Université de Tours-Centre National de la Recherche Scientifique (CNRS)

Subjects
viscosity solutions, capacity potential, 01 natural sciences, Physics::Fluid Dynamics, Bernoulli's principle, gradient flow, Mathematics - Analysis of PDEs, Inviscid flow, Free boundary problem, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, Mathematics - Optimization and Control, minimizing movement, Bernoulli problem, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, free boundary problem, 010101 applied mathematics, Flow (mathematics), non local geometric evolution, Optimization and Control (math.OC), shape optimization, front propagation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Balanced flow, Viscosity solution, Energy (signal processing), Analysis of PDEs (math.AP)
Abstract

International audience; In Cardaliaguet-Ley (2006) we have defined a viscosity solution for the gradient flow of the exterior Bernoulli free boundary problem. We prove here that the associated energy is non decreasing along the flow. This justifies the "gradient flow" approach for such kind of problem. The proof relies on the construction of a discrete gradient flow in the flavour of Almgren-Taylor-Wang (1993) and on proving it converges to the viscosity solution.

Published
2006
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