Dynamics and interactions of free fluid interfaces

本项目的主要研究目标是：了解近平涡片的长时间动力学；解析确定涡旋片双向模式系统中孤波的稳定性；并研究带电流体之间的界面行为。第一个项目的基础是证明线性化涡片问题的色散估计以及非线性问题的先验能量估计。最终目的是证明，如果初始位形接近平坦，则完整非线性运动方程的解在时间上全局存在。第二个领域的重点是理解和规避技术障碍，这些障碍是在没有假设存在首选传播方向的情况下推导的自由流体界面近似模型中试图建立孤立波的轨道稳定性时出现的。单向性在稳定性分析中起着重要而微妙的作用，推广到双向系统已被证明是困难的。第三个项目的重点是了解外加电场如何加剧/阻止界面破裂。电磁效应引入了新的非局部和非线性效应，使系统的分析复杂化。预测两种相互剪切的流体之间的界面如何随时间移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情况中：在海洋表面，在大气层之间，在船的尾迹或飞机的机翼后面。一个很自然的问题是，一个最初非常接近完美平面的界面是否会随着时间的推移而变得完美平面。在本研究计划的第一部分中，将对接近平面的界面进行非常精确的定量描述。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：它们可以传播很远的距离，基本上不变。孤立波是如此稳定，这一点可以很好地理解为它们进化的各种模型，其中一个假设是波有一个首选的运动方向。这个项目的这一部分在没有这种假设的情况下研究了它们的稳定性，同时也研究了海底地形可能（可能）干扰这种波的方式。本项目的最后一部分涉及开发和分析模拟外加电磁场对流体界面影响的系统。以这种方式控制流体界面引起了人们极大的兴趣。应用包括：动态波导和透镜，高速开关，涂层和冷却过程。

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.