Some Dynamical Problems in Fluids and Plasmas

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This project studies dynamical structures in collisionless plasmas, incompressible fluids, and surface water waves. Stability of a variety of coherent structures including steady galaxies, periodic and solitary water waves, and electromagnetic plasma equilibria will be further explored. Some of these structures are unstable, which means that a small perturbation creates a large disturbance. The goal is to determine which structures are stable and which are unstable. Another focus is to understand what role these steady structures play in the long time dynamics. The invariant structures near unstable steady states, including unstable (stable) manifolds and heteroclinic orbits, will be investigated. Particular emphasis will be on the inviscid limit of these dynamical structures, which is important for understanding the transition to turbulence and the observed structures. Nonlinear damping near certain stable equilibria of non-dissipative fluids and plasmas will also be studied.The general goal of this research is to develop new and effective mathematical methods for the analysis of dynamic behavior in fluids and plasmas. Mathematical advances here will improve our understanding of the basic physical mechanisms, and also can lead to better numerical methods for simulations of these complex phenomena. Plasma is a gas composed of particles, each carrying a positive or negative electrical charge. Most of the universe is plasma; examples include the solar wind, the ionosphere, galactic nebulae, and comet tails. Plasmas also arise in physics and engineering, for instance in nuclear fusion. The stability and dynamical structures of fluids, plasmas and water waves are relevant for such diverse applications as fusion energy research, satellite and radio communications in space, wave motion in the atmosphere and oceans, and turbulent behavior of fluids. The mechanisms of nonlinear damping in the non-dissipative models are important for understanding how the large scale coherent structures (such as galaxies) in the fluid and plasma turbulence develop and evolve.