Congested Aggregation via Newtonian Interaction
Congested Aggregation via Newtonian Interaction
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DOI：
10.1007/s00205-017-1156-6
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发表时间：
2018-01-01
影响因子：
2.5
通讯作者：
Yao, Yao
中科院分区：
文献类型：
--
作者：
Craig, Katy;Kim, Inwon;Yao, Yao
我们考虑了一个拥挤的聚集模型，该模型通过非局部牛顿引力和硬高度约束的竞争效应来描述密度的演变。这与现有的关于排斥-吸引非局部相互作用模型的文献相反，其中排斥效应是由相互作用核或扩散的增加引起的。我们将我们的模型表述为相互作用能量的瓦瑟斯坦梯度流，并对密度的高度施加惩罚。从这个角度看，问题可以看作是具有退化扩散的Keller-Segel方程的奇异极限。两个关键性质将我们的问题与先前关于高度约束方程的工作区分开来：相互作用核的非凸性（这将模型置于经典梯度流理论的范围之外）和速度场对密度的非局部依赖性（这导致问题缺乏比较原理）。为了克服这些障碍，我们将非凸能量梯度流的最新研究结果与粘滞解理论相结合。我们用Hele-Shaw型自由边界问题描述了贴片解的动力学特性，并利用这一特性表明，在二维空间中，贴片解在长时间极限下收敛于一个圆盘的特征函数，具有明确的能量衰减速率。我们认为，当前工作的一个关键贡献是我们的混合方法，将能量方法与粘度解理论相结合。
We consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal Newtonian attraction and a hard height constraint. This provides a counterpoint to existing literature on repulsive-attractive nonlocal interaction models, where the repulsive effects instead arise from an interaction kernel or the addition of diffusion. We formulate our model as the Wasserstein gradient flow of an interaction energy, with a penalization to enforce the constraint on the height of the density. From this perspective, the problem can be seen as a singular limit of the Keller-Segel equation with degenerate diffusion. Two key properties distinguish our problem from previous work on height constrained equations: nonconvexity of the interaction kernel (which places the model outside the scope of classical gradient flow theory) and nonlocal dependence of the velocity field on the density (which causes the problem to lack a comparison principle). To overcome these obstacles, we combine recent results on gradient flows of nonconvex energies with viscosity solution theory. We characterize the dynamics of patch solutions in terms of a Hele-Shaw type free boundary problem and, using this characterization, show that in two dimensions patch solutions converge to a characteristic function of a disk in the long-time limit, with an explicit rate on the decay of the energy. We believe that a key contribution of the present work is our blended approach, combining energy methods with viscosity solution theory.