Mathematical Sciences: Singularities in Vortical Flows and Dispersive Systems

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9623087 Caflisch This proposal is for development of analytical and computational methods for describing singularities in the solutions of partial differential equations (PDEs), and for application of those methods to problems of fluid mechanics and other physical systems. The key step in the analysis is transformation of the variables of the PDE to obtain an "unfolded" system, the solutions of which are non-singular. This will be used for vortex sheets, swirling flow, magnetohydrodynamics and crystal patterns. For these problems we will attempt to find singular solutions and to classify their generic type. This proposal is for research on "singularities" in physics and engineering problems. Singularities are points at which the measurable quantities in the system change abruptly or become very large. An important example, which is a main focus of this project is the development of very large rotational velocity in a fluid such as air or liquid. These singularities are believed to be a primary cause of turbulence, an important but poorly understood phenomena in many applications. The goals of this research project are to find these singularities, classify their possible behavior and use them to gain understanding of turbulence and other problems.