Geometric gradient-flow dynamics with singular solutions
Adaptation and Self-Organizing Systems
2008-04-28 v2 Pattern Formation and Solitons
Abstract
The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.
Cite
@article{arxiv.0704.2369,
title = {Geometric gradient-flow dynamics with singular solutions},
author = {Darryl D. Holm and Vakhtang Putkaradze and Cesare Tronci},
journal= {arXiv preprint arXiv:0704.2369},
year = {2008}
}