Mathematical Sciences: Nonlinear Reaction-Diffusion

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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.