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arXiv · 0704.2369

Geometric gradient-flow dynamics with singular solutions

Darryl D. Holm·Vakhtang Putkaradze·Cesare Tronci
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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.

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Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. 2008-04-28. Geometric gradient-flow dynamics with singular solutions. https://doi.org/10.1016/j.physd.2008.04.010

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