Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach

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 $\epsilon$ and the time scale $\tau$. When the ratio $\frac{\epsilon }{\tau }$ 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 $\epsilon \to 0$, 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.

References in corpus (3)

• Viscous corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems
• Hemihelical local minimizers in prestrained elastic bi-strips
• Chirality transitions in frustrated ferromagnetic spin chains: a link with the gradient theory of phase transitions

Cited by in corpus (1)

• On the approximation of quasistatic evolutions for the debonding of a thin film via vanishing inertia and viscosity