论文标题
通过倾斜来探索能量散落景观的家庭 - 三种类型的EDP收敛
Exploring families of energy-dissipation landscapes via tilting -- three types of EDP convergence
论文作者
论文摘要
我们介绍了梯度系统融合的两个新概念$（\ Mathbf Q，\ Mathcal E_ \ Varepsilon，\ Mathcal R_ \ Varepsilon）$ to（\ Mathbf Q，\ Mathbf Q，\ Mathcal E_0，\ Mathcal R_0）$。这些新概念称为“用倾斜的EDP收敛”和“与倾斜的接触”。两者都是基于梯度系统溶液的能量分离原理（EDP）制定的，并且可以看作是Sandier和Sanfaty首先引入的梯度流的伽马连接的改进。构建了两个新概念，以避免有时会作为EDP连接的限制出现的“不自然”限制梯度结构。 EDP连接与倾斜相连是通过$（\ Mathbf Q，\ Mathcal e_ \ varepsilon，\ Mathcal R_ \ varepsilon）$的完整“倾斜”副本的完整家族来加强EDP连接的增强。它避免了不自然的限制梯度结构，但是根据这个概念，许多有趣的系统是非构造的。接触 - 倾斜和倾斜的EDP收敛是倾斜的EDP收敛性的放松，并且仍然避免了不自然的极限，但适用于更广泛的序列$（\ Mathbf Q，\ Mathcal E_ \ Varepsilon，\ Mathcal R_ \ varepsilon）$。在本文中，我们定义了这些概念，研究它们的特性，并将它们与经典的EDP收敛联系起来。我们说明了许多测试问题的不同概念。
We introduce two new concepts of convergence of gradient systems $(\mathbf Q, \mathcal E_\varepsilon,\mathcal R_\varepsilon)$ to a limiting gradient system $(\mathbf Q, \mathcal E_0,\mathcal R_0)$. These new concepts are called `EDP convergence with tilting' and `contact--EDP convergence with tilting'. Both are based on the Energy-Dissipation-Principle (EDP) formulation of solutions of gradient systems, and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the `unnatural' limiting gradient structures that sometimes arise as limits in EDP-convergence. EDP-convergence with tilting is a strengthening of EDP-convergence by requiring EDP-convergence for a full family of `tilted' copies of $(\mathbf Q, \mathcal E_\varepsilon,\mathcal R_\varepsilon)$. It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact--EDP convergence with tilting is a relaxation of EDP convergence with tilting, and still avoids unnatural limits but applies to a broader class of sequences $(\mathbf Q, \mathcal E_\varepsilon,\mathcal R_\varepsilon)$. In this paper we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.
