Mathematical Sciences: Nonlinear Reaction-Diffusion

研究人员研究了化学工程中出现的两个反应扩散问题和气候学中的一个问题，每个问题都显示了一个自由或移动的边界。第一个问题涉及反应器或催化剂颗粒中的分数级放热反应。然后可能存在一个死核(反应物浓度为零的区域)，其边界随时间演变。特别注意一种被称为阴影系统的高化学扩散率近似。第二个问题涉及流体和固体之间的相互作用，这是化学和材料工程中经常出现的现象。研究了多孔固体燃烧的各种性质，如转化时间、转化前沿和穿透前沿。在比先前考虑的更一般的设置中检验了所谓的伪稳态近似的有效性。气候学问题涉及冰盖区域对能量平衡方程中出现的太阳常数和红外辐射常数等参数的依赖关系。这项研究涉及反应扩散系统的各个方面。这个名字源于一些问题，比如煤的燃烧，因为氧气在热煤中扩散，并与之反应，将其转化为产品。相应的数学模型也出现在其他环境中，但作了一些修改：例如，在催化转化器中，在生态竞争中，在污染物的传播和清除中，在脉冲沿神经的传播中，在气候模拟中。研究人员的兴趣在于数学框架和具体应用。利用最近开发的技术，他致力于获得对物理感兴趣的数量的估计，例如在气固反应中固体转化所需的时间，并发现大气中温室气体数量的变化对地球冰盖的影响。

The investigator studies two reaction-diffusion problems arising in chemical engineering and one from climatology, each exhibiting a free or moving boundary. The first problem concerns an exothermic reaction of fractional order in a reactor or catalyst particle. There may then exist a dead core (a region where the reactant concentration vanishes) whose boundary evolves in time. Particular attention is paid to a high chemical diffusivity approximation known as the shadow system. The second problem treats the interaction between fluid and solid phases, a frequent phenomenon in chemical and materials engineering. Various properties of the combustion of a porous solid are studied, such as the conversion time, the conversion front, and the penetration front. The validity of the so-called pseudo-steady-state approximation is examined in a more general setting than previously considered. The climatology problem concerns the dependence of ice-covered regions on parameters such as the solar constant and the infrared radiation constants that appear in the energy balance equation. This study deals with aspects of reaction-diffusion systems. The name originated with problems such as the combustion of coal as oxygen diffuses through the hot coal and reacts with it, converting it to products. The corresponding mathematical model also appears, with some modifications, in other settings: for instance, in catalytic converters, in ecological competition, in the spreading of contaminants and in their removal, in propagation of impluses along nerves, and in climate modeling. The investigator's interest lies both in the mathematical framework and in specific applications. Using techniques developed recently, he directs his efforts to obtaining estimates for quantities of physical interest such as the time required for conversion of the solid in a gas-solid reaction and to finding the effect on the earth's icecaps of a change in the amount of greenhouse gases in the atmosphere.