Rate-dependent gradient flows of integral currents via varicurrents

Abstract

Motivated by the analysis of recent models of dislocation motion in crystalline materials, this work introduces space-time varicurrents, a form of Young measures over integral currents, to describe their evolutions with rate-dependent dissipation potentials. This addresses the problem that rate-dependent versions of the (rate-independent) space-time variation suffer from a lack of lower semicontinuity with respect to the natural weak topology. This causes a dynamical relaxation to occur by means of infinitely fine oscillations. Varicurrents retain a description of this ``microstructure'' (in space-time) and, based on this theory, we are able to establish the existence of solutions to a rate-dependent gradient flow. This is in contrast to the classical formulation using integral currents, which does not have a solution in general.