Gamma-convergence of gradient flows with applications to Ginzburg-Landau

2004 ◽  
Vol 57 (12) ◽  
pp. 1627-1672 ◽  
Author(s):  
Etienne Sandier ◽  
Sylvia Serfaty
Keyword(s):  
Gamma Convergence ◽  
Gradient Flows ◽  
Ginzburg Landau

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 Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

2021 ◽  
Author(s):  
Alexander Mielke ◽  
Alberto Montefusco ◽  
Mark A. Peletier
Keyword(s):  
Energy Dissipation ◽  
Test Problems ◽  
Gradient System ◽  
Gamma Convergence ◽  
Gradient Flows ◽  
Gradient Systems ◽  
New Concepts

AbstractWe introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ ( Q , E 0 , R 0 ) . These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

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