Nonlocal Diffusion Problems
Nonlocal Diffusion Problems
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DOI：
10.1090/surv/165
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发表时间：
2010-10
期刊：
影响因子：
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通讯作者：
F. Andreu-Vaillo;J. Mazón;J. Rossi;J. J. Toledo-Melero-J.
中科院分区：
文献类型：
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作者：
F. Andreu-Vaillo;J. Mazón;J. Rossi;J. J. Toledo-Melero-J.
非局部扩散问题出现在各种各样的应用中，包括生物学、图像处理、粒子系统、凝聚模型和数学金融。这些类型的问题也是非常感兴趣的纯数学内容。这本书介绍了最近的结果与不同的边界条件的非局部发展方程，从线性理论和移动到非线性的情况下，包括两个非局部模型的演变沙堆。解的存在性和唯一性，以及它们的渐近行为。此外，作者提出的结果有关的限制的解决方案的非局部方程作为一个重标度参数趋于零。这些限制程序最常用的扩散模型恢复：热方程，$p$-拉普拉斯演化方程，多孔介质方程，总变流量，对流扩散方程和局部模型的演变沙堆由于Aronsson-Evans-Wu和Prigozhin。假定读者熟悉泛函分析和偏微分方程的基本概念和技术。文本是独立的，否则与论述强调直观的理解和结果与充分的证据。它适合研究生或研究人员。作者涵盖了一个近年来受到极大关注的主题。这本书的目的是作为一个参考工具，为一般观众在分析和偏微分方程，包括数学家，工程师，物理学家，生物学家，和其他感兴趣的非局部扩散问题。
Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content. This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the $p$-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin. Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers. The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.