论文标题
与速率无关的系统和梯度系统有关，以实现单均匀电位的情况
Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials
论文作者
论文摘要
我们考虑了希尔伯特空间上的一个非负和单均能量功能$ \ Mathcal J $。本文提供了相关梯度流程方程的解决方案与通过速率涉及速率的系统生成的能量解决方案之间的确切关系，该系统以时间依赖性功能$ \ MATHCAL E（T，U）= T \ MATHCAL J（u）$以及标准作为散散距离。两种流之间的关系是通过依赖解决方案的时间依赖性的重新培训来给出的，这可以从两个方程式中的能量和耗散的均匀性中猜测出来。我们提供了几个示例，包括总变化流量，并表明这两个系统通过依赖于时间的解决方案的对等效度。在数学上使关系严格包括对能量解决方案中的跳跃的仔细分析，这些跳动对应于梯度流动方程的溶质素的恒定间隔。作为主要结果，我们获得了非依赖速率的系统的非平凡的存在和独特性。
We consider a non-negative and one-homogeneous energy functional $\mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $\mathcal E(t,u)=t \mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.
