scholarly journals Kurdyka–Łojasiewicz–Simon inequality for gradient flows in metric spaces

2019 ◽  
Vol 372 (7) ◽  
pp. 4917-4976 ◽  
Author(s):  
Daniel Hauer ◽  
José M. Mazón
Keyword(s):  
Metric Spaces ◽  
Gradient Flows

Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

2019 ◽  
Vol 25 ◽  
pp. 28
Author(s):  
Florentine Fleißner
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Gradient Flow ◽  
Gradient Flows ◽  
Flow Motion ◽  
Limit Functional ◽  
Γ Convergence

We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Γ-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.

scholarly journals The Systems of Nonlinear Gradient Flows on Metric Spaces and Its Gamma-Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Mao-Sheng Chang ◽  
Bo-Cheng Lu
Keyword(s):  
Nonlinear System ◽  
Metric Spaces ◽  
Gradient Flow ◽  
Gamma Convergence ◽  
Gradient Flows ◽  
Energy Functionals ◽  
Several Variables ◽  
Flow Systems

We first establish the explicit structure of nonlinear gradient flow systems on metric spaces and then develop Gamma-convergence of the systems of nonlinear gradient flows, which is a scheme meant to ensure that if a family of energy functionals of several variables depending on a parameter Gamma-converges, then the solutions to the associated systems of gradient flows converge as well. This scheme is a nonlinear system edition of the notion initiated by Sylvia Serfaty in 2011.

scholarly journals Weighted Energy-Dissipation principle for gradient flows in metric spaces

2019 ◽  
Vol 127 ◽  
pp. 1-66 ◽  
Author(s):  
Riccarda Rossi ◽  
Giuseppe Savaré ◽  
Antonio Segatti ◽  
Ulisse Stefanelli
Keyword(s):  
Energy Dissipation ◽  
Metric Spaces ◽  
Gradient Flows ◽  
Weighted Energy

scholarly journals Variational Convergence of Gradient Flows and Rate-Independent Evolutions in Metric Spaces

2012 ◽  
Vol 80 (2) ◽  
pp. 381-410 ◽  
Author(s):  
Alexander Mielke ◽  
Riccarda Rossi ◽  
Giuseppe Savaré
Keyword(s):  
Metric Spaces ◽  
Variational Convergence ◽  
Gradient Flows

scholarly journals Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons

2022 ◽  
Vol 29 (2) ◽  
Author(s):  
Manuel Friedrich ◽  
Lennart Machill
Keyword(s):  
Metric Spaces ◽  
Three Dimensional ◽  
Gradient Flows ◽  
Dimensional Model ◽  
Discrete Approximations ◽  
Three Dimensional Model ◽  
One Dimensional ◽  
Von Karman ◽  
Von Kármán ◽  
Appl Math

AbstractWe consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin’s-Voigt’s rheology derived from a three-dimensional model at a finite-strain setting in Friedrich and Kružík (Arch Ration Mech Anal 238: 489–540, 2020). As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004) and complement the $$\Gamma $$ Γ -convergence analysis of elastic von Kármán ribbons in Freddi et al. (Meccanica 53:659–670, 2018). Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.

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