POWRE: Research in Applied Analysis and Conservation Laws

POWRE：应用分析和守恒定律研究

• 批准号: 9973475
• 负责人: Barbara Keyfitz
• 金额: $ 3.57万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973475&HistoricalAwards=false
• 关键词: POWRE; Research; Applied; Analysis; Conservation

The PI will use this grant as partial salary support for a sabbatical leave at Brown University, visiting mathematicians in the Mathematics and Applied Mathematics Departments.Research and scholarly activities planned for the period include preparing a monograph on nonstrictly hyperbolic conservation laws and conservation laws that change type. The proposer has worked extensively on problems in this area since 1976. Recognition of the possible utility of these models in engineering applications, along with continuing mathematical advances, has inspired this effort to bring together in one place the results of the proposer and of other researchers in the field, to broaden the scope of examples treated, and to make the subject accessible to practitioners in other fields. A major contribution of the book will be to introduce to mathematicians interested in the theory and numerical analysis of conservation lawsa number of interesting new prototype problems and applications. Some of the motivation for studying nonstrictly hyperbolic problems comes from elasticity and continuum mechanics. Other application areas that have furnished examples include flow through porous media (petroleum reservoirs) and multiphase flows (for example, mixtures of liquid and steam, bubbly liquids, and fluidized bed reactors) which are important in several fields of engineering and technology.Keyfitz also proposes to advance her recent research on multidimensional conservation laws, in cooperation with Dafermos, Strauss, and other Brown faculty. The lack of a good existence theory in more than one space dimension is the outstanding theoretical problem in quasilinear hyperbolic equations. One approach is through the study of self-similar problems. Results obtained so far by Canic and Keyfitz indicate that some basic questions in both elliptic and hyperbolic equations must be answered. Many aspects of nonlinear analysis may play a role here. Brown's mathematics department offers great strength in this area,both in its permanent faculty and in the visitors who have planned to be there in 1999-2000.In addition to the theoretical importance of existence questions, there is much applications interest in self-similar problems, which include such benchmark flows as shock reflection by a wedge.The POWRE grant offers the proposer focussed support for research and scholarship at a critical time, when she has been able to conclude some administrative and family responsibilities. It will also allow her to learn new methodologies (in nonlinear analysis) and to engage in cross-disciplinary study. The outcome, both in the form of the eventual book and in research advances in multidimensional conservation laws, will mark a significant increase in the proposer's research strength and visibility.

PI 将利用这笔补助金作为布朗大学学术休假的部分工资支持，访问数学和应用数学系的数学家。该期间计划的研究和学术活动包括编写一本关于非严格双曲守恒定律和变化守恒定律的专着类型。 自 1976 年以来，提议者对这一领域的问题进行了广泛的研究。对这些模型在工程应用中可能的实用性的认识，以及不断的数学进步，激发了这一努力，将提议者和其他研究人员的结果汇集到一个地方在该领域，扩大所处理实例的范围，并使其他领域的从业者能够理解该主题。 本书的一个主要贡献是向对守恒定律理论和数值分析感兴趣的数学家介绍一些有趣的新原型问题和应用。 研究非严格双曲问题的一些动机来自弹性和连续介质力学。 提供示例的其他应用领域包括穿过多孔介质（石油储层）的流动和多相流（例如液体和蒸汽的混合物、气泡液体和流化床反应器），这些在工程和技术的多个领域中都很重要。Keyfitz 还建议与达弗莫斯、施特劳斯和其他布朗大学教师合作，推进她最近对多维守恒定律的研究。 在多维空间中缺乏良好的存在理论是拟线性双曲方程中突出的理论问题。 一种方法是通过研究自相似问题。 Canic 和 Keyfitz 迄今为止获得的结果表明，椭圆方程和双曲方程中的一些基本问题都必须得到解答。 非线性分析的许多方面可能在这里发挥作用。 布朗大学数学系在这一领域具有强大的实力，无论是在其常任教职人员还是计划在 1999-2000 年访问该系的访客方面。除了存在性问题的理论重要性之外，人们对自相似问题也有很多应用兴趣，其中包括诸如楔子冲击反射等基准流量。POWRE 赠款为提议者在关键时刻（当她能够承担一些行政和家庭责任时）提供对研究和学术的集中支持。 它还将使她能够学习新的方法（非线性分析）并从事跨学科研究。 其成果，无论是最终书籍的形式还是多维守恒定律的研究进展，都将标志着提案者的研究实力和知名度的显着提高。