Interaction of Particles with Non-central Potential: Gradient Flows and Singular Solutions for Evolution of Geometric Continuum Quantities

Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces. Several examples are given. The motivating example is the directed self assembly of a large number of particles for technological purposes such as nano-science processes, in which the particle interactions are anisotropic. This application leads to the derivation and analysis of gradient flow equations on Lie algebra valued densities. The Riemannian structure of these gradient flow equations is also discussed.