POWRE: Research in Applied Analysis and Conservation Laws

PI将使用这笔赠款作为在布朗大学休假的部分工资支持，访问数学和应用数学系的数学家。在此期间计划的研究和学术活动包括准备一本关于非严格双曲型守恒定律和变型守恒定律的专著。自1976年以来，该提出者在这一领域的问题上做了大量工作。对这些模型在工程应用中可能有用的认识，以及不断的数学进步，促使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，扩大所处理的例子的范围，并使其他领域的从业人员能够接触到这一主题。这本书的一个主要贡献将是向对守恒律的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于弹性力学和连续介质力学。其他已经提供了例子的应用领域包括流经多孔介质(石油储集层)和多相流(例如，液体和蒸汽的混合物、起泡液体和沸腾床反应器)，这些在几个工程和技术领域都很重要。Keyfitz还建议与Dafermos、Strauss和其他Brown教授合作，推进她最近对多维守恒定律的研究。在多维空间中缺乏一个好的存在性理论是拟线性双曲型方程的一个突出的理论问题。一种方法是通过研究自相似问题。到目前为止，Canic和Keyfitz得到的结果表明，椭圆型方程和双曲型方程中的一些基本问题都必须得到回答。非线性分析的许多方面可能在这里发挥作用。布朗的数学系在这一领域提供了很大的优势，无论是在它的常设教员和计划在1999-2000年来那里的访客。除了存在问题的理论重要性之外，还有许多应用程序对自相似问题感兴趣，其中包括像楔子冲击反射这样的基准流。POWRE补助金为提名人在关键时刻提供集中支持研究和学术研究，当她已经能够完成一些行政和家庭责任。它还将使她能够学习新的方法(在非线性分析中)，并从事跨学科研究。无论是以最终的书的形式，还是在多维守恒定律的研究进展中，结果都将标志着提出者的研究实力和知名度的显著增加。

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.