Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics
Congming Li Eric S. Wright
Communications on Pure & Applied Analysis 2002, 1(1): 77-84 doi: 10.3934/cpaa.2002.1.77
The carbonate system is an important reaction system in natural waters because it plays the role of a buffer, regulating the pH of the water. We present a global existence result for a system of partial differential equations that can be used to model the combined dynamics of diffusion, advection, and the reaction kinetics of the carbonate system.
keywords: partial differential equations Carbonate system
Modeling chemical reactions in rivers: A three component reaction
Congming Li Eric S. Wright
Discrete & Continuous Dynamical Systems - A 2001, 7(2): 377-384 doi: 10.3934/dcds.2001.7.373
What follows is the analysis of a model for dynamics of chemical reactions in a river. Dominant forces to be considered include diffusion, advection, and rates of creation or destruction of participating species (due to chemical reactions). In light of this, the model we will be using will be based upon a nonlinear system of reaction-advection-diffusion equations. The nonlinearity comes solely from the influences of the chemical reactions.
First, we will establish some general results for reaction diffusion systems. In particular, we will illustrate a class of reaction diffusion systems whose solutions are bounded from below by zero. We will also provide a local existence result for this class of problems. Afterwards, we will focus on the dynamics of an equidiffusive three component reaction system. Specifically, we will provide conditions under which one could be guaranteed the existence of global solutions. We will also discuss the qualities of the $\omega$-limit set for this system.
keywords: Reaction diffusion systems local existence.
Macrotransport in nonlinear, reactive, shear flows
Eric S. Wright
Communications on Pure & Applied Analysis 2012, 11(5): 2125-2146 doi: 10.3934/cpaa.2012.11.2125
In 1953, G.I. Taylor published his paper concerning the transport of a contaminant in a fluid flowing through a narrow tube. He demonstrated that the transverse variations in the fluid's velocity field and the transverse diffusion of the solute interact to yield an effective longitudinal mixing mechanism for the transverse average of the solute. This mechanism has been dubbed ``Taylor Dispersion.'' Since then, many related studies have surfaced. However, few of these addressed the effects of nonlinear chemical reactions upon a system of solutes undergoing Taylor Dispersion. In this paper, I present a mathematical model for the evolution of the transverse averages of reacting solutes in a fluid flowing down a pipe of arbitrary cross-section. The technique for deriving the model is a generalization of an approach by introduced by P.C. Fife. The key outcome is that while one still finds an effective mechanism for longitudinal mixing, there is also a effective mechanism for nonlinear advection.
keywords: Partial differential equations asymptotic analysis Taylor dispersion reaction-diffusion. macrotransport
On a nonlinear parabolic system-modeling chemical reactions in rivers
Wenxiong Chen Congming Li Eric S. Wright
Communications on Pure & Applied Analysis 2005, 4(4): 889-899 doi: 10.3934/cpaa.2005.4.889
We study the global existence and qualitative properties of the solutions of nonlinear parabolic systems. Such systems commonly arise in situations pertaining to reactive transport. Particular examples include the modeling of chemical reactions in rivers or in blood streams.
In this paper, we first establish a maximum principle that generates a-priori bounds for solutions to a broad class of parabolic systems. Afterward, we develop an alternative technique for establishing global bounds on solutions to a specific system of three equations that belong to a different class of parabolic systems. Finally, we prove that the only bounded traveling wave solutions to this system are constants.
keywords: reaction-diffusion systems heat kernel Parabolic PDE maximum principle

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