Calculus of Variations and Geometric Measure Theory
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G. Scilla - F. Solombrino

Delayed loss of stability in singularly perturbed finite-dimensional gradient flows

created by solombrin on 04 Sep 2017
modified on 05 Sep 2017

[BibTeX]

Submitted Paper

Inserted: 4 sep 2017
Last Updated: 5 sep 2017

Year: 2017

ArXiv: 1709.00737 PDF

Abstract:

In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Hilbert space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time $t^*$ of the limit. For our class of functionals, $t^*$ coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time $t_c$ where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of $t^*$, rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini, where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.

Keywords: singular perturbations, Gradient Flow, Variational methods, heteroclinic solutions, dynamical systems


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