Qualitative Properties of Stochastic Partial Differential Equations

随机偏微分方程的定性性质

• 批准号: 1102646
• 负责人: Carl Mueller
• 金额: $ 30万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1102646&HistoricalAwards=false
• 关键词: Qualitative; Properties; Stochastic; Partial; Differential

This proposal deals with three topics in stochastic partial differential equations (SPDE). The first question deals with uniqueness. Unless SPDE have unique solutions, they are useless for modeling. Recent work by the proposer, L. Mytnik, and E. Perkins settled a longstanding question about uniqueness of SPDE related to the heat equation. Using these newly developed tools, the proposer will attack uniqueness questions for other types of equations, and study uniqueness among nonnegative solutions. The second question deals with traveling waves, which occur widely in physical systems. Recently the proposer, L. Mytnik, and J. Quastel have solved a problem raised by Brunet and Derrida, involving the asymptotic speed of traveling waves when the noise term is small. The proposer will would like to extend this analysis to other equations, and related particle systems. The third topic involves stochastic wave equations. The most commonly studied SPDE are variants of the heat equation, but others such as the stochastic wave equation are receiving increasing attention. With D. Geba, the proposer will like to study stochastic wave equations involving null forms. Using some function spaces introduced by Bourgain, the proposer's goal is to study short-time existence. The ideas developed should be relevant to other classes of equations. The field of SPDE is rapidly expanding. As technology moves towards the micro level, the effect of random noise becomes more and more important. Since the most important mathematical modeling tools we have are ordinary and partial differential equations, the study of SPDE is becoming essential in many applied fields. Even though SPDE is now several decades old, the area has only recently received widespread attention. There is still a need for pioneering work to establish a toolbox for the area. This proposal deals with three types of problems in SPDE. The proposer believes that through the study of such specific examples is the right way to generate methods and further our understanding, for both theory and applications. In summary, the proposer believes that SPDE will play an essential role in future applications of mathematics. He wishes to develop the theory and help to train graduate students so that this area can fulfill its potential.

该提案涉及随机部分微分方程（SPDE）中的三个主题。 第一个问题涉及独特性。 除非SPDE具有独特的解决方案，否则它们对于建模没有用。 提议者L. Mytnik和E. Perkins的最新工作解决了一个长期存在的问题，该问题关于SPDE与热方程相关的独特性。 提议者使用这些新开发的工具，将为其他类型的方程式发动独特性问题，并研究非负解决方案的独特性。 第二个问题涉及在物理系统中广泛发生的行驶波。 最近，提议者L. Mytnik和J. Quastel解决了Brunet和Derrida提出的问题，涉及噪声项很小时行驶波的渐近速度。 建议者希望将此分析扩展到其他方程式和相关的粒子系统。 第三个主题涉及随机波方程。 最常见的SPDE是热方程式的变体，但是其他诸如随机波方程的其他方程正在受到越来越多的关注。 借助D. Geba，建议者将希望研究涉及无效形式的随机波程。 使用Bourgain引入的一些功能空间，提案者的目标是研究短期存在。 提出的想法应与其他等式类别相关。 SPDE领域正在迅速扩展。 随着技术朝着微观层面发展，随机噪声的效果变得越来越重要。 由于我们拥有的最重要的数学建模工具是普通和部分微分方程，因此SPDE的研究在许多应用领域中变得至关重要。 即使SPDE现在已经有几十年了，该地区直到最近才受到广泛关注。 仍然需要开拓工作来为该区域建立工具箱。 该建议涉及SPDE中的三种类型的问题。 提议者认为，通过研究此类示例是生成方法的正确方法，并进一步了解理论和应用。 总而言之，该提议者认为，SPDE将在数学的未来应用中发挥重要作用。 他希望发展理论并帮助培训研究生，以便该领域能够发挥其潜力。