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首页> 外文期刊>Jahresbericht der Deutschen Mathematiker-Vereinigung >Self-similarity in Smoluchowski's Coagulation Equation
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Self-similarity in Smoluchowski's Coagulation Equation

机译：Smoluchowski凝聚方程的自相似性

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摘要

Smoluchowski's coagulation equation is one of the fundamental deterministic models that describe mass aggregation phenomena. A key question in the analysis of this equation is whether the large-time behaviour of solutions is universal and described by special self-similar solutions. This issue is however only well-understood for the small class of solvable kernels while the analysis for non-solvable kernels still poses many challenging problems. Our main focus in this article will be to describe recent progress in the analysis of self-similar solutions of Smoluchowski's equation for non-solvable kernels. Existence results for self-similar solutions with finite mass have been available for some time for a large class of kernels. In contrast, the uniqueness of such solutions has been an open problem for some time. A first uniqueness result has recently been obtained for kernels that are in a certain sense close to constant. We present here a shorter proof under an additional assumption on the kernels that makes the analysis significantly simpler. We also give an overview of recent results on the existence of fat tail solutions for non-solvable kernels.
机译：Smoluchowski的凝聚方程是描述质量聚集现象的基本确定性模型之一。分析该方程的一个关键问题是溶液的长时间行为是否具有普遍性，并由特殊的自相似解来描述。但是，对于一小类可解决的内核，只有很好地理解了此问题，而对非可解决的内核的分析仍然存在许多具有挑战性的问题。本文的主要重点是描述非可溶核的Smoluchowski方程的自相似解的分析的最新进展。具有有限质量的自相似解的存在结果已经有一段时间可以用于大量的内核了。相反，这种解决方案的独特性在一段时间以来一直是一个未解决的问题。最近从某种意义上接近常数的内核获得了第一唯一性结果。在对内核进行额外假设的情况下，我们在此提供了一个较短的证明，这使得分析变得更加简单。我们还概述了有关非可解内核的胖尾解的最新结果。

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