POWRE: Research in Applied Analysis and Conservation Laws

PI将使用这笔补助金作为部分工资支持，以在布朗大学休假，访问数学系和应用数学系的数学家。这一时期计划的研究和学术活动包括准备一本关于非严格双曲守恒定律和改变类型的守恒定律的专著。自1976年以来，提案人对这一领域的问题进行了广泛的研究。认识到这些模型在工程应用中的可能效用，以及持续的数学进步，激发了这一努力，将提出者和该领域其他研究人员的结果集中在一个地方，扩大了所处理的例子的范围，并使其他领域的实践者能够接触到这一主题。本书的一个主要贡献将是向数学家介绍对守恒定律的理论和数值分析感兴趣的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力学和连续介质力学。其他已提供示例的应用领域包括通过多孔介质（油藏）的流动和多相流动（例如，液体和蒸汽的混合物，气泡液体和流化床反应器），它们在几个工程和技术领域中都很重要。基菲茨还提议与达福莫斯、施特劳斯和其他布朗大学教员合作，推进她最近对多维守恒定律的研究。缺乏一个好的多维空间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迄今所得到的结果表明椭圆型方程和双曲型方程的一些基本问题必须得到解答。非线性分析的许多方面都可能在这里发挥作用。布朗大学的数学系在这一领域具有强大的实力，无论是在其永久教员还是计划在1999-2000年期间来这里的访客。除了存在性问题的理论重要性外，自相似问题也有很多应用兴趣，其中包括楔形激波反射等基准流。在申请人能够完成一些行政和家庭责任的关键时期，POWRE资助将为研究和奖学金提供重点支持。这也将使她学习新的方法（非线性分析），并从事跨学科的研究。结果，无论是最终的书的形式，还是多维守恒定律的研究进展，都将标志着提案人的研究实力和知名度的显著提高。

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.