Long-Time Behavior in Free-Surface Problems in Fluid Dynamics

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The motion of fluids with free surfaces occurs in many problems in engineering and science; the mathematical theory of such flows is only partially developed, however. While smooth flows for short times have recently become fairly well understood, there is little theory developed for other situations. This project explores models for interfacial Navier-Stokes flows after the onset of topological transitions, global existence of small solutions for vortex sheets, and the mathematical theory of non-Newtonian Hele-Shaw flow, among other problems.There are many possible applications of the theory of free-surface flows in fluid dynamics in science and engineering, such as the drilling and cementing of oil wells, understanding blood flow, or understanding the mixing of fluids in the atmosphere and ocean, as well as problems in microfluidics. In general terms, the proposed research will improve the understanding of wave breaking, fluid mixing, turbulence, and stability of such flows. The investigator will utilize techniques of both mathematical analysis and scientific computing, using insight gained from each approach to more efficiently and more effectively use the other.