偏微分方程和粒子系统中的自由边界问题
Free boundary problems in PDEs and particle systems
摘要：
在这卷理论模型的运输中存在的一个自由边界的发展。宏观输运规律用偏微分方程描述。当系统是开放的，有几个机制来耦合系统与外力。这里考虑一类系统，其中与外部的相互作用发生在对应的自由边界。分析了具有相同结构的连续模型和离散模型。在第一部分中，详细地讨论了与Stefan问题有关的一个自由边界问题。对于该模型，提出了一个新的松弛解的概念，并证明了它的全局存在性和唯一性。它也表明，这是流体动力学极限的经验质量密度的相关粒子系统。第二部分讨论了其他几种模式。期望的是，基本模型证明的结果扩展到这些其他情况。所有的模型讨论在这卷有兴趣的问题出现在几个研究领域，如热传导，排队论，火的传播，界面动力学，人口动力学，进化的生物系统与选择mechanism.In一般的研究人员感兴趣的偏微分方程和随机过程之间的关系可以找到在这卷的扩展，这对应于现代数学物理。
英文摘要：
In this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDEs. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analyzed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases. All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms.In general researchers interested in the relations between PDEs and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics.
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