Dynamics and interactions of free fluid interfaces

该项目的主要研究目标是：了解近平涡面的长时间动力学;分析确定涡面双向模型系统中孤立波的稳定性;以及研究带电流体之间的界面行为。 第一个项目是植根于证明线性涡面问题的色散估计沿着与非线性问题的先验能量估计。最终的目标是表明，如果初始配置是近平坦的，那么完整的非线性运动方程的解决方案存在于全球的时间。第二个领域的重点是理解和规避技术障碍时，试图建立孤立波的近似模型中的自由流体界面的轨道稳定性推导没有假设，有一个优选的传播方向。单向性在稳定性分析中起着重要但微妙的作用，并且已经证明难以推广到双向系统。第三个项目是围绕理解如何施加电场可以加剧/逮捕界面破裂。 电磁学引入了新的非局部和非线性效应，使系统的分析变得复杂。预测相互剪切的两种流体之间的界面如何随时间运动是数学流体力学的一个中心问题。这种非常普遍的情况发生在许多实际感兴趣的情况下：在海洋表面，在大气层之间，在船的尾流或飞机机翼后面。一个自然的问题是，一个界面最初非常接近于完全平坦，随着时间的推移，它实际上是否会变得完全平坦。 在本研究项目的第一部分，将开发非常精确的定量描述近平界面。 该项目的第二个组成部分涉及孤立水波的演变，当它们发生在开阔的海洋中时，被称为海啸。 这些波的破坏力与它们很难被破坏这一事实直接相关：它们可以在基本不变的情况下传播很长的距离。 孤立波是如此稳定，这是很好地理解了各种模型的演变，其中一个假设，波有一个首选的运动方向。 本项目的这一部分在没有这种假设的情况下研究它们的稳定性，并研究海底地形（可能）干扰这种波的方式。 这个项目的最后一部分是关于开发和分析系统，模拟应用电磁场对流体界面的影响。 以这种方式操纵流体界面引起了很大的兴趣。 应用包括：动态波导和透镜，高速开关，涂层和冷却工艺。

The primary research objectives of this project are: to understand the long time dynamics of nearly flat vortex sheets; to determine analytically the stability of solitary waves in bi-directional model systems of vortex sheets; and to study the behavior of interfaces between electrified fluids. The first project is rooted in proving dispersive estimates for the linearized vortex sheet problem along with a priori energy estimates for the nonlinear problem. The ultimate goal is to show that if the initial configuration is nearly flat then solutions of the full nonlinear equations of motion exist globally in time. The second area is focused on understanding and circumventing technical obstructions which arise when trying to establish orbital stability for solitary waves in approximate models for free fluid interfaces derived without the assumption that there is a preferred direction of propagation. Uni-directionality plays an important but subtle role in the stability analysis and generalization to bi-directional systems has proven difficult. The third project is centered around understanding how an applied electric field can exacerbate/arrest rupture of the interface. The electro-magnetics introduce new nonlocal and nonlinear effects which complicate the analysis of the system.Predicting how an interface between two fluids which are shearing past one another moves in time is a central problem in mathematical hydrodynamics. This very general scenario occurs in many situations of practical interest: on the surface of the ocean, between layers in the atmosphere, in the wake of a boat or behind the wing of an aircraft.A natural question to ask is whether or not an interface initially very close to being perfectly flat will in fact become perfectly flat as time evolves. In the first part of this research project, very precise quantitative descriptions of nearly flat interfaces will be developed. The second component of this project concerns the evolution of solitary water waves, which are called tsunamis when they occur in the open ocean. The destructive power of these waves is directly related to the fact that they are very difficult to disrupt: they can travel great distances essentially unchanged. That solitary waves are so stable is well understood for a variety of models for their evolution wherein one makes an assumption that the wave has a preferred direction of motion. This part of this project studies their stability without this assumption, and also to examine ways in which bottom topography can (possibly) disturb such waves. The final part of this project is concerned with developing and analyzing systems which model the effects of an applied electromagnetic field on a fluid interface. There is a great deal of interest in manipulated fluid interfaces in this way. Applications include: dynamic wave guides and lenses, high speed switching, coating and cooling processes.