Research in Nonlinear Problems Arising in Mechanics

Slemrod在研究气体、流体和颗粒材料的本构关系的多尺度模型时使用的科学工具是这些材料的动力学理论 奇异摄动极限这些奇异摄动极限使他能够在大的空间和时间尺度上恢复模型。然而，统一模型中大尺度和小尺度的联系将通过Slemrod开发的新技术来完成，并受到物理学家在研究临界现象时所使用的程序的激励。Slemrod称这种方法为“广义有理逼近”。它提供了一个稳定的方式外推从小尺度到大尺度的材料模型。在研究液体、气体和颗粒物质的动力学时，重要的是要有良好的数学模型来帮助预测物理现象。同样重要的是，这些模型要在整个长度和时间尺度范围内有效。例如，远离飞机或航天飞机重新进入地球大气层的大气（稀薄气体）的运动使用的大尺度模型不同于用于模拟飞机或航天飞机附近激波层的小尺度动力学的模型。 然而，为了计算效率，一个统一的模型有效的大范围的长度和时间尺度是可取的。 Slemrod在他的研究中将制定一种方法来寻找流体，气体和颗粒材料的这种模型。

The scientific tools used by Slemrod in his research on findingmulti-scale models for constitutive relations for gases, fluids, and granular materials are kinetic theories for these materials and singular perturbation limits. These singular perturbation limits allow him to recover models on large space and time scales. However the linking of the large and small scales in unified models will be accomplished by a new technique developed by Slemrod and motivated by the procedure used by physicists in the study of critical phenomena. Slemrod calls the method" generalized rational approximation". It provides a stable way of extrapolating from small scale to large scale models of materials. In the study of the dynamics of liquids, gases, and granular materials it is important to have good mathematical models to aid in the prediction of physical phenomena. It is also important to have these models to be valid on a whole range of length and time scales. For example the motion of the atmosphere (a rarefied gas) far away from a plane or a space shuttle re-entering the earth's atmosphere uses a large scale model different than that used to model the small scale dynamics of a shock layer near the plane or shuttle. However for computational efficiency one unified model valid on a large range of length and time scales is desirable. Slemrod in his research will formulate a method for finding such models for fluids, gases, and granular materials.