U.S.-Austria Cooperative Research: Volterra Integral Equations and Applications

该奖项将支持一项为期两年的数学合作研究工作，合作伙伴包括南伊利诺伊大学卡本代尔分校的Ronald C.Grimmer教授、爱荷华州立大学的Richard K.Miller教授、奥地利格拉茨大学的Wilhelm Schappacher教授和同样在Graz大学的Wolfgang Desch博士。合作的数学家们正在研究涉及Banach空间中所谓的Volterra积分微分方程的一些相关问题。他们的研究将处理有关偏微分积分方程解的控制和定性行为的各种问题。特别令人感兴趣的是确定在初始或边界数据中具有奇异性的解的行为。重要的是要确定这些奇点是否传播，如果是的话，另外还要确定这些奇点的行为。还将包括对解的渐近行为的研究，特别是与控制问题相关的解的渐近行为。这项研究的预期结果将在很大程度上是抽象的数学性质，但这项工作很可能导致适用于实际问题的数学技术，例如在研究同时具有粘性和弹性性质的材料的物理行为时。这种粘弹性介质的例子有聚合物固体、高相对分子质量聚合物流体、血液和流态化煤。这里所研究的方程的另一个应用是电动力学。在这个项目中合作的数学家具有相似和互补的优势和兴趣。他们过去的合作卓有成效，此次获奖将使他们能够继续开展互利合作。

This award will support a two-year collaborative research effort in mathematics involving Professor Ronald C. Grimmer, Southern Illinois University at Carbondale; Professor Richard K. Miller, Iowa State University; Professor Wilhelm Schappacher, University of Graz, Austria; and Dr. Wolfgang Desch, also at the University of Graz. The cooperating mathematicians are studying a number of related problems involving so-called Volterra integrodifferential equations in Banach space. Their research will deal with various questions concerning control and qualitative behavior of solutions of partial differential integral equations. Of particular interest will be determining the behavior of solutions which have singularities in the initial or boundary data. It is important to determine if these singularities propagate and, if so, additionally to determine the behavior of these singularities. Included also will be the study of asymptotic behavior of solutions especially as it is related to control problems. The expected results of this research will be largely of an abstract mathematical nature, but the work may well lead to mathematical techniques which are applicable to practical problems, as for example in the study of the physical behavior of materials having both viscous and elastic properties. Examples of such viscoelastic media are polymeric solids, high molecular weight polymer fluids, blood, and fluidized coal. Another application of the equations being studied here is to electrodynamics. The mathematicians collaborating in this project have both similar and complementary strengths and interests. They have worked together productively in the past, and this award will permit them to continue their mutually beneficial cooperation.