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The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcys 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.
Time-dependent Hamiltonian dynamics is derived for a curve (molecular strand) in $mathbb{R}^3$ that experiences both nonlocal (for example, electrostatic) and elastic interactions. The dynamical equations in the symmetry-reduced variables are written
We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the non-local e
We prove the existence of weak solutions to a system of two diffusion equations that are coupled by a pointwise volume constraint. The time evolution is given by gradient dynamics for a free energy functional. Our primary example is a model for the d
Cycling is a green transportation mode, and is promoted by many governments to mitigate traffic congestion. However, studies concerning the traffic dynamics of bicycle flow are very limited. This study experimentally investigated bicycle flow dynamic
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - mbox{div}[mathbb{A}(x,t,u, abla u)]= mbox{div}[{mathbf F}]$ with homogeneous Dirichlet boundary conditions