Fluid Flows at Large

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Fluid Flows at Large Abstract of Proposed ResearchMaria E. Schonbek This project is to continue work on the analysis of three fundamental equations that are basic for fluid mechanics. These are the Navier-Stokes and Magneto-hydrodynamic (MHD) equations of hydrodynamics and the 2d quasi-geostrophic (2dQG) equation of meteorology. For the Navier-Stokes equation, our efforts will center on analyzing the self-similar solutions on a half-space or, possibly, a conical domain. Interest will be on their regularity and describing the solutions explicitly. For the MHD equations, research will center on the asymptotic behavior of solutions when dissipation is solely dependent on velocity. For the geo-strophic equations, the research will focus on describing the long-time behavior of flows around obstacles. Analysis of the Navier-Stokes equations is fundamental for understanding fluid flows. Any explicit solution helps in describing what possible flows can be sustained. The MHD equations arise when one adds the effect of a magnetic field and the fluid is assumed to contain electrically charged particles and ions. This project will investigate the solutions of these equations in the absence of magnetic dissipation. The analysis of the QG equations is a model for air-flow around an obstacle; a common problem in meteorology.