标题： 几何梯度流动力学中的奇异解 显示英文标题

标题： Geometric gradient-flow dynamics with singular solutions

摘要： 任意几何量的梯度流动力学是通过Darcy定律的推广推导出来的。 我们考虑拉格朗日和欧拉公式中的流动。 拉格朗日公式包括流体动力学的耗散修正。 显示标量、1-形式和2-形式自组织的欧拉方程可简化为非局部特征方程。 我们确定这些方程的奇异解，对应于坍缩（聚集）状态，并讨论它们的演化。 显示英文摘要

摘要： 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.