Hyperbolic-Parabolic Balance Laws with Applications

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This project aims to study a general system of partial differential equations called hyperbolic-parabolic balance laws, with two specific applications in mind: one in bio-medicine and another in gas dynamics. There is a variety of gas-dynamic phenomena that can be described mathematically by hyperbolic-parabolic balance laws. The swirl of complicated gas flows surrounding a space vehicle reentering the Earth's atmosphere is probably one of the most spectacular examples for which these balance laws serve as a (simplified) mathematical description. The second application motivating this research is chemotaxis, the movement of micro-organisms in response to a chemical stimulus. The mechanism of chemotaxis is ubiquitous in biology and medicine, from migration of bacteria or leukocytes to cancer metastasis. Arising from the physical world, the parabolic-hyperbolic systems do not fit exactly into the traditional classification in the theory of partial different equations, and the solutions' behavior is governed by both hyperbolicity and parabolicity, plus a chemical reaction. Integrated into the research activities of the project there is an educational component. It includes curriculum development of hand-on modeling of real-world problems by partial differential equations, one-on-one research mentorship to undergraduate students, and direct participation of research by Ph.D. students. The project encourages the participation of students at all levels, especially students from under-represented groups.The project has two principal mathematical objectives. The first objective is to study the stability of equilibrium solutions. This is to understand how an initial perturbation, such as inaccuracy in measurement, propagates into the solution. The project team is particularly interested in the wave pattern formed by perturbation at long time. The research unifies the theories on hyperbolic balance laws and hyperbolic-parabolic conservation laws in this regard. It has direct application to gas flows in translational and vibrational non-equilibrium. The second objective is to study another model system, Keller-Segel-Fisher/KPP system, beyond the application of the above general theory. It is a chemotaxis model that has logarithmic sensitivity and logistic growth. Here the logarithmic sensitivity is to account for Fechner's law: Subjective sensation is proportional to the logarithm of the stimulus intensity. An important practical problem is what happens if the chemical signal is near void at one end of the domain. This is equivalent to the logarithmic function is near its singular point. In the project, we carry out an in-depth investigation into such a situation. The goals are existence of solution global in time, large time behavior, and possibly large data solutions or strong wave solutions.This project is jointly funded by the Applied Mathematics Program of the Division of Mathematical Sciences and by the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.