Research in Nonlinear Problems Arising in Mechanics

Slemrod在研究寻找气体、流体和颗粒材料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和奇异摄动极限。这些奇异扰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连接将由Slemrod开发的一种新技术完成，并受到物理学家在研究临界现象时使用的程序的激励。Slemrod称这种方法为“广义有理近似”。它提供了一种从小尺度到大尺度材料模型的稳定外推方法。在研究液体、气体和颗粒状物质的动力学时，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，让这些模型在整个长度和时间尺度范围内有效。例如，远离飞机或重新进入地球大气层的航天飞机的大气（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的小尺度激波层动力学的模型。然而，为了提高计算效率，需要一个在大范围长度和时间尺度上有效的统一模型。Slemrod将在他的研究中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.