Nonlinear Stability and Instability of Fluid and Plasma Equilibria

研究了流体和等离子体中平衡态的稳定性和不稳定性，重点研究了流体中的二维欧拉方程以及等离子体物理和恒星动力学中的Vlasov系统。他的目标是发展线性增长模式的一般方法(特别是对于非对称模式和边界模式)，在尖锐的线性稳定性准则下研究非线性稳定性，以及研究通常的能量-Casimir方法不能处理的非单调平衡，以及当粒子轨迹不可积时从线性不稳定建立非线性不稳定。等离子体是一种由粒子组成的气体，每个粒子都带有正电荷或负电荷，每个电荷的浓度大致相等。宇宙的大部分是等离子体；例如太阳风、电离层、银河星云和彗星尾巴。等离子体也出现在物理和工程中，例如在核聚变中。研究人员研究流体和等离子体中平衡的稳定性和不稳定性。这里的数学进步加深了我们对支配稳定和不稳定的物理机制的理解，因此可以带来更好的数值方法来模拟这些复杂的现象。流体稳定性对于理解海浪和大气运动的机制以及在工程应用中都是重要的。等离子体稳定性的问题出现在各种各样的领域，如太阳黑子、基础等离子体物理学、聚变机的设计和等离子体显示设备。

The investigator studies the stability and instability ofequilibria in fluids and plasmas, focusing on the two-dimensionalEuler equation in fluids and on Vlasov systems in plasma physicsand stellar dynamics. He aims to develop general methods offinding linearly growing modes (especially for asymmetric modesand boundary modes), to investigate nonlinear stability undersharp linear stability criteria and for non-monotonic equilibriathat cannot be handled by the usual energy-Casimir method, and toestablish nonlinear instability from linear instability when theparticle trajectory is non-integrable. A plasma is a gas composed of particles, each carrying apositive or negative electrical charge, with approximately equalconcentrations of each 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 investigator studies thestability and instability of equilibria in fluids and plasmas. Mathematical advances here enhance our understanding of thephysical mechanisms that govern stability and instability, and socan lead to better numerical methods for simulations of thesecomplex phenomena. Fluid stability is important in understandingthe mechanisms of ocean waves and the movements of the atmosphereas well as in engineering applications. Questions of plasmastability arise in areas as diverse as sunspots, fundamentalplasma physics, the design of fusion machines, and plasma displaydevices.