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Your search keyword '"Minimizing movement"' showing total 4 results
Start Over Descriptor "Minimizing movement" Topic mathematics - analysis of pdes
4 results on '"Minimizing movement"'

1. Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

Author

Anna Kubin, Daniele De Gennaro, and De Gennaro, Daniele

Subjects
Mathematics - Differential Geometry, Minimizing movement, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), 49M25, Applied Mathematics, 49Q20, FOS: Mathematics, [MATH] Mathematics [math], Analysis, Geometric evolution equations, 53C24, Analysis of PDEs (math.AP)
Abstract

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided., 38 pages, 2 figure. arXiv admin note: text overlap with arXiv:2004.04799 by other authors

Published
2022

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. The $L^1$ gradient flow of a generalized scale invariant Willmore energy for radially non increasing functions

Author

Dayrens, Fran��ois, Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Willmore energy, gradient flow, Mathematics - Analysis of PDEs, Mathematics - Classical Analysis and ODEs, radial functions, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 49J45 49J52 46S30, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], minimizing movement, Analysis of PDEs (math.AP)
Abstract

We use the minimizing movement theory to study the gradient flow associated with a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, one can define a Willmore energy on regular functions of L 1 (R d). This functional is extended to every L 1 function by taking its lower semi-continuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions, i.e. functions with balls as level sets. In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then we show that the flow consists on an erosion of the initial data. The erosion speed is given by a first order ordinary equation.

Published
2014

4. 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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