Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach. (5th October 2018)

• Record Type: Journal Article
• Authors: Scilla, Giovanni; Solombrino, Francesco
• Is Part Of: Nonlinearity. Volume 31:Number 11(2018:Nov.)
• Page Start: 5036
• Page End: 5074
• Publication Date: 2018-10-05
• Subjects: gradient flow -- Balanced Viscosity solutions -- singular perturbations -- minimizing movements scheme -- variational methods -- discrete interfacial energy; 34K26 -- 34K18 -- 34C37 -- 47J35 -- 49J45; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aad6ac
• Languages: English
• ISSNs: 0951-7715

Abstract

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 ε and the time scale τ . When the ratio diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as a formal limit, when, 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.