Dispersive PDE and Interfacial Fluid Dynamics

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The Principal Investigator and his collaborators will address several problems in interfacial fluid dynamics and dispersive partial differential equations.The following are the specific activities to be undertaken: (A) Time-periodic interfacial fluid flows, such as water waves and vortex sheets with surface tension, will be investigated, both computationally and analytically. (B) Breakdown of smooth solutions for interfacial flows will be investigated, specifically addressing the question of whether a Moore singularity can occur for flows with surface tension. (C) Well-posedness of rotational interfacial Hele-Shaw flows will be proved. (D) Rigorous studies of the well-posedness or ill-posedness of equations with degenerate dispersion, such as the Rosenau-Hyman equations, will be performed.There are many practical situations in science and engineering which involve interfaces between moving fluids, such as the mixing of fluids in the ocean, or the motion of contaminants in groundwater. The research to be undertaken in this project will provide insight into the predictability of such motions, as well as phenomena such as wave breaking and fluid mixing. The investigator will use methods of both mathematical analysis and scientific computing as appropriate in carrying out the research. Both undergraduate and graduate students will participate in the research.