This proposal addresses rigorous analysis of several free boundary problems arising in fluid dynamics of Hele-Shaw flow and dendritic growth. These problems have been investigated for decades by applied mathematician and engineers; so there are many numerical and formal asymptotic results. Some of the results appear to be contradictory and there are controversies in the literature so there is a need for mathematically rigorous results. The Saffman-Taylor instability and Mullins-Sekerka instability have long been standard examples of pattern formation and instabilities in non equilibrium systems. In these problems, when surface tension is zero, solutions exist for a continuum set of values of some parameter (finger width in viscous fingering or crystalline anisotropy in dendritic growth). When the surface tension is nonzero and small, however, the Selection Principle states that solutions exist only for certain discrete values of the parameter. In this project we intend to provide a theoretical underpinning for the selection principle in problems such as the Saffman-Taylor bubble in a Hele-Shaw cell and the rising inviscid bubble in a vertical tube. The stability analysis of these problems will also be investigated by developing new mathematical methods, including results from complex analysis, nonlinear integro-differential equations, nonlinear partial deferential equations and exponential asymptotics.
The principal investigator is currently affiliated with Morgan State University, a state university with historically predominant percentage of African American students. The results obtained in this project during the next three years will be used for the development of undergraduate courses in Complex variable and applications, Ordinary and partial differential equations. These are core courses for the mathematics department at Morgan State University. African American senior undergraduate and graduate students (both male and female) will be engaged in this project.
