权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Long-time behaviour for kinetic models of clustering and non-local diffusion equations

聚类和非局部扩散方程动力学模型的长期行为

基本信息

批准号：
396845724
负责人：
Dr. Sebastian Throm
金额：
--
依托单位：
Institut für Analysis und Numerik
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/396845724?language=en
关键词：
Long time behaviour kinetic models

项目摘要

Agglomeration and clustering are quite universal phenomena which appear in a broad range of different natural and industrial processes. Some typical examples are the formation of rain drops, the creation of polymers, formation of planets but also biological processes such as the creation of so-called marine snow. The common feature of all these phenomena, which occur on a very broad range of length scales, is the formation of larger clusters due to the coalescence with smaller particles. One of the fundamental problems in this context, which is also the main goal of this project, consists in understanding the long-time behaviour of this coarsening process as precisely as possible. One of the most important mathematical models for the description of the phenomena above is Smoluchowski's coagulation equation. For the latter, numerical simulations suggest that the long-time behaviour is self-similar. With the exception of very few special cases, for which explicit solution formulas exist, this so-called scaling hypothesis has not yet been proven. The main concern of this project is thus the proof of this conjecture for certain selected models for which no explicit solutions can be given. Moreover, the long-time behaviour of a simplified model for agglomeration is to be studied, where a target cluster moves through a random background of particles and coalesces with them upon touching. The expected evolution of the size of the target cluster for large times is again self-similar. Supplementary, non-local diffusion equations are to be considered as they arise on the one hand naturally in the consideration of coagulation models while on the other hand their study exhibits similar difficulties.

集聚和集群是一种相当普遍的现象，出现在各种不同的自然和工业过程中。一些典型的例子是雨滴的形成，聚合物的产生，行星的形成，以及生物过程，如所谓的海洋雪的产生。所有这些现象的共同特征是，由于与较小颗粒的聚结而形成较大的团簇，这些现象发生在非常宽的长度尺度范围内。在这种情况下，这也是本项目的主要目标的基本问题之一，在于尽可能精确地了解这种粗化过程的长期行为。描述上述现象的最重要的数学模型之一是Smoluchowski的凝结方程。对于后者，数值模拟表明，长期行为是自相似的。除了极少数的特殊情况，其中明确的解决方案的公式存在，这个所谓的缩放假设尚未得到证明。因此，这个项目的主要关注点是证明这个猜想的某些选定的模型，没有明确的解决方案可以给出。此外，还将研究一个简化的凝聚模型的长期行为，其中一个目标团簇在随机的颗粒背景中移动，并在接触时与它们合并。大时间的目标集群的大小的预期演变再次是自相似的。补充，非局部扩散方程被认为是因为它们出现在一方面自然在考虑凝血模型，而另一方面，他们的研究表现出类似的困难。