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.
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BibTeX
@misc{cesik2026ratedependent,
title = {{Rate-dependent gradient flows of integral currents via varicurrents}},
author = {Antonín Češík and Malte Kampschulte and Filip Rindler},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05005},
eprint = {2611.05005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05005},
}
RIS
TY - UNPB
TI - Rate-dependent gradient flows of integral currents via varicurrents
AU - Antonín Češík
AU - Malte Kampschulte
AU - Filip Rindler
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05005
UR - https://arxiv.org/abs/2611.05005
AB - 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.
ER -