POWRE: Research in Applied Analysis and Conservation Laws

PI将使用此补助金作为部分工资支持在布朗大学休假，访问数学和应用数学系的数学家。计划在此期间进行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改变类型的守恒定律的专著。 自1976年以来，提案人就这一领域的问题开展了广泛的工作。 认识到这些模型在工程应用中的可能效用，沿着不断的数学进步，激发了这一努力，将提议者和该领域其他研究人员的结果汇集在一个地方，以扩大所处理的例子的范围，并使其他领域的从业者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定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弹性力学和连续介质力学。 其他应用领域提供的例子包括通过多孔介质（石油储层）和多相流（例如，液体和蒸汽，气泡液体的混合物，流化床反应器），这是在工程和技术的几个领域的重要流动。Keyfitz还提出了推进她最近的多维守恒定律的研究，与Dafermos，施特劳斯和其他布朗教师合作。 拟线性双曲型方程在多维空间中缺乏一个好的解的存在性理论是其突出的理论问题。 一种方法是通过研究自相似问题。 到目前为止，Canic和Keyfitz的结果表明，一些基本的问题，在椭圆和双曲方程必须回答。 非线性分析的许多方面都可以在这里发挥作用。 布朗大学的数学系在这一领域提供了很大的优势，无论是在其永久的教师和游客谁计划在1999- 2000年在那里。除了存在问题的理论重要性，有很多应用兴趣在自相似问题，其中包括楔体激波反射等基准流。POWRE赠款为研究和奖学金的申请人提供重点支持在关键时刻，她已经能够完成一些行政和家庭责任。 这也将使她学习新的方法（在非线性分析），并从事跨学科研究。 结果，无论是在最终的书的形式和在多维守恒定律的研究进展，将标志着在提议者的研究实力和知名度显着增加。

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.