Well-Posedness and Singularity Formation in Applied Free Boundary Problems

This project concerns the modelling of fluid motion as motivated by applications of dielectric fluids in microfluidic devices, the combustion and motion of flame fronts, and waves in water. The emphasis is in understanding for how long the mathematical models used to study such phenomena remain valid and predictive, in various physical regimes. For dielectric fluids, the focus is on situations where the density of electrical charge changes rapidly over small regions. For combustion and flame fronts, the aim is to study a hierarchy of mathematical models and determine how well the simpler models serve to approximate behavior in the more complicated models. While for water waves, the goal is to consider how wavetrains with different frequencies interact, or how wavetrains interact with certain kinds of bottom topography. The project provides research training opportunities for graduate students. The project will analyze the Melcher-Taylor leaky dielectric model, with the goal of establishing local well-posedness theory, study a mechanism for shock formation via analytical and numerical approaches, and consider global existence for the Kuramoto-Sivashinsky equation in more than one spatial dimension. Validation theorems relating the Kuramoto-Sivashinsky equation, coordinate-free models of flame fronts, and hydrodynamic flame models will be also investigated. Furthermore, the investigator will establish local well-posedness for water waves with spatially quasiperiodic initial data, and analyze related models, such as the Benjamin-Ono equation or the Euler equations for interfacial flows, in the spatially quasiperiodic setting. 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.