Propagation Phenomena in Reaction-advection-diffusion Equations PDF Download

Author: Christopher Henderson
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Languages : en
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Book Description
Reaction-advection-diffusion (RAD) equations are a class of non-linear parabolic equations which are used to model a diverse range of biological, physical, and chemical phenomena. Originally introduced in the early twentieth century as a model for population dynamics, they have been used in recent years in diverse contexts including climate change, criminal behavior, and combustion. These equations are characterized by the combination of three behaviors: spreading, stirring, and growth/decay. The main focus of mathematical research into RAD equations over the past century has been in characterizing the propagation of solutions. Indeed, these equations are characterized by the invasion of an unstable state by a stable state at a constant rate (for instance, the invasion of empty space by a population until the environmental carrying capacity is reached). In general, this can be characterized by the existence, uniqueness, and stability of traveling wave solutions, or solutions with a fixed profile which move at a constant speed in time. In general, the speed and shape of these traveling waves gives us the speed with which the stable state invades the unstable state. This thesis assumes the following trajectory, investigating two specific RAD equations: the Fisher-KPP equation, used in population dynamics, and a coupled reactive-Boussinesq system, used to model combustion in a fluid. For the former equation, we prove results regarding the precise spreading rate, and for the latter equation, we prove an existence result for a special solution that generalizes the traveling wave. In the first part of this thesis, we prove two results quantifying the precise speed of spreading for solutions to the Cauchy problem of the Fisher-KPP equation. The first of these results, concerning localized initial data, provides intuition for a lower order term obtained non-rigorously in. Specifically, we prove a quantitative convergence-to-equilibrium result in a related model, which has been used as a close approximation of the Fisher-KPP equation. The second of these results, concerning non-localized initial data and building on the work of Hamel and Roques, quantifies the super-linear in time spreading of the population. Here we compute the highest order term in the spreading for a broad class of initial data. In the second part of this thesis, we look at a coupled system that models combustion in a fluid, and we prove a qualitative propagation result. Unlike classical models, this relatively new system accounts for the effect of advection induced by the buoyancy force that results from the evolution of the temperature. Essentially, this means that we take into account the phenomenon that ``hot air rises.'' We exhibit a generalized traveling wave solution of this system, called a pulsating front, in two-dimensional periodic domains. To our knowledge, this is the first result regarding the existence of ``pulsating fronts'' in a coupled system.