Research in Nonlinear Problems Arising in Mechanics

斯莱姆罗德在研究寻找气体、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。这些奇异的微扰极限使他能够恢复大空间和时间尺度上的模型。然而，大尺度和小尺度在统一模型中的连接将由Slemrod开发的一种新技术来完成，该技术的动机是物理学家在研究临界现象时所使用的程序。斯莱姆罗德称这种方法为“广义有理逼近”。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、气体和颗粒物质的动力学时，重要的是要有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(一种稀薄气体)使用的大尺度模型不同于用来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，为了提高计算效率，需要一个在大范围的长度和时间范围内有效的统一模型。斯莱姆罗德在他的研究中将制定一种方法，为流体、气体和颗粒材料寻找这样的模型。

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.