Motion of Interface Between Two Fluids

建议：dms-9801094首席研究员：吴思觉摘要：吴思觉将研究水波的长时间行为，以及一般两层流体的运动。为了了解水波的长时间行为，吴想要确定这种波的自由面是否保持非自交，以及水波问题的解是否始终存在，因为水波的初始自由面是静止水的小扰动。更准确地说，她想描述那些保证水波自由表面始终不自交的初始自由表面的特征。她还想研究那些不是一直存在的水波问题解的爆破机制。解决这些问题的方法源于调和分析和偏微分方程组理论。它们包括：衰减估计，最大原理论证，以及问题的自相似解的构造。为了理解任意两个叠加流体之间的界面运动，吴打算研究一些基本问题，如运动方程的解的存在性以及解的奇性轮廓。为了实现这一目标，她希望采用她早期研究水波的方法，推导出一个等同于描述界面运动的方程的拟线性方程。采用这种方法的原因是，模拟双流体界面运动的方程是高度非线性的。一般来说，人们对拟线性方程的理解比对完全非线性方程的理解更好。建议的研究是由Nalimov，Yosihara和Craig开始的，后来由首席研究员提出的关于水波问题解的存在性和唯一性的工作，以及Sulem，Dochun和Robert，Caflisch和Orellana，以及Ebin关于涡片运动适定性的工作的继续。流体波以各种形式出现在日常生活中最熟悉的一些体验中，从巨大噪音对我们耳膜的刺耳冲击到海滩上舒缓的潮起潮落。与波动相关的丰富多样的现象为几代物理学家和数学家提供了一个重要而具有挑战性的研究课题。两种流体的自由界面运动的一般问题可应用于各种具体的物理问题。它被用来理解湍流模型中的流体混合、边界层分离、声音的产生和相干结构。这项拟议工作的长期目标是了解表面波中的波浪破碎和涡旋片中的奇异性机制，这两个主题都具有重大的物理和工程意义。

Proposal: DMS-9801094 Principal Investigator: Sijue Wu Abstract: Wu will study the long time behavior of water waves, as well as the motion of a general two-layered fluid flow. To understand the long time behavior of a water wave, Wu would like to determine whether the free surface of such a wave remains non-self-intersecting and whether solutions of water wave problems exist for all time, given that the initial free surface of the water wave is a small perturbation of still water. To be more precise, she would like to characterize those initial free surfaces that guarantee the non-self-intersection of the free surface of a water wave for all time. She also wants to study the blow-up mechanism for those solutions of the water wave problem that fail to exist for all time. The methods used to tackle these problems arise from harmonic analysis and the theory of partial differential equations. They include: decay estimates, maximal principle arguments, and the construction of a self-similar solution to the problem. To understand the motion of the interface between any two superposed fluids, Wu intends to study such fundamental questions as the existence of solutions to the equation governing the motion and the singularity profiles of solutions. To achieve this goal, she hopes to adapt the methods in her earlier work on water waves to derive a quasilinear equation equivalent to the equation that describes the motion of the interface. The reason for this approach is that the equation which models the motion of a two-fluid interface is highly nonlinear. In general, one has a better understanding of quasilinear equations than of fully nonlinear equations. The proposed research is a continuation of the work started by Nalimov, Yosihara and Craig and later advanced by the principal investigator on the existence and uniqueness of solutions of water wave problems and the work of Sulem, Dochun and Robert, Caflisch and Orellana, and Ebin on the well-posedness of vortex sheet motion. Fluid waves, in numerous guises, are present in some of the most familiar experiences of daily life, from the jarring impact of loud noises on our eardrums to the soothing ebb and flow of surf at a beach. The rich variety of phenomena associated with wave motion have provided generations of physicists and mathematicians with an important and challenging research subject. The general problem of motion of the free interface of two superposed fluids has applications to a wide variety of concrete physical problems. It has been used to understand the mixing of fluids, the separation of boundary layers, the generation of sounds, and coherent structures in models of turbulence. The long term objective of the proposed work is to understand wave-breaking in surface waves and the singularity mechanism in vortex sheets, both topics that have significant ramifications for physics and engineering.