POWRE: Research in Applied Analysis and Conservation Laws

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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.