Mathematical Sciences: A Study of Weak Solutions for the Euler Equations and Related Equations from Plasma Physics

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9307726 Majda There are many unresolved fundamental problems about weak solutions of the incompressible Euler equations, such as vortex sheets, and regularizations of these weak solutions by smoothing the initial data or adding viscosity. Furthermore, since these problems are extremely challenging ones to try to solve with numerical methods, it is natural to first consider related simpler model problems. Recent research shows that the solutions of the two-component Vlasov Poisson equations (2CVPE) for a collisionless plasma of electrons and positively charged ions, with the initial condition being an appropriate measure, have many properties that are direct analogues of solutions of the 2-D vorticity equation for an incompressible fluid with vortex sheet initial data. The 2CVPE is simpler to treat analytically and easier to solve numerically. It should be an extremely useful and an important model problem for suggesting answers to the open problems about vortex sheets. The investigator conducts a numerical investigation of the 2CVPE and its "viscous" regularization, the two-component Fokker-Planc equation. The goal is to determine properties of solutions to these systems of equations when the initial data is a measure. Recently discovered exact solutions of the 2CVPE will focus this work. The investigator also conducts a numerical investigation of the 2-D incompressible Euler and Navier-Stokes equations with appropriate vortex sheet initial data. He uses the insight gained from the study of related model problems from plasma physics to motivate and guide this work. This work is designed to provide answers to many of the open problems about properties of solutions of the Euler equations with vortex sheet initial data. Fluid dynamics is the study of continuous media like air and water. Scientists formulate problems in this field by applying the fundamental laws of physics such as the conservation of mass, momentum and energy. For most probl ems this procedure leads to sets of mathematical equations that are too complicated to solve. Based on physical or mathematical intuition, the scientist then tries to find a simplified formulation of the original problem that is simple enough to solve, but sufficiently complicated to describe the essential features of the original problem. The mixing process between layers of fluid that have different densities, like air and water, or layers of fluid moving at different velocities, is very complicated and occurs in a wide variety of important applications. The flow of air over an airplane is also very complicated. In order to predict the trailing wake behind an airplane, or the mixing process in a fluid, scientists often use a simplified model of the fluid called a vortex sheet. This is the simplest realistic description of not only these problems, but many other large Reynold's number flows. Unfortunately, despite the wide-spread use of vortex sheet models in many problems in fluid dynamics, many mathematical properties of vortex sheets remain poorly understood. The goal of this proposal is to provide answers to many of the fundamental questions about the mathematical properties of vortex sheets. Successful completion of the proposed work will lead to an increased understanding of the strengths and limitations when using vortex sheets to model complicated fluid flows.