2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159747The 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.28 pages, 1 figure, to appear on Physica DAdaptation and Self-Organizing SystemsPattern Formation and Solitonstext