Analysis of moving interface problems in fluid dynamics

理想流体流动用流体力学欧拉方程建模；它们是（3+1）维时空中非线性守恒定律的耦合系统，是所有流体运动模型的基础。包含声理论的可压缩流表现出不连续的波剖面；例子包括激波、接触不连续面、移动的材料界面、移动的真空边界、熵波和各种附加的不连续波模式。对这种多维波型的分析，特别是对传播至少一个不连续面的欧拉方程的解的分析，是理解基本物理现象的基础。本研究工作的目标是将这种多维不连续波轮廓视为移动自由边界问题，并为双曲和退化双曲系统的适定性发展一种理论，构造具有有限时间奇点的解，其中传播的超表面相互碰撞，研究奇异渐近极限，如消失粘度极限和零表面张力极限，并针对相变模型中普遍存在的所谓双曲抛物型问题，提出了一种新的非线性稳定性理论。具有移动界面的多相流体流动在许多物理和工程应用中发挥着核心作用，从海风吹过海洋表面产生的飓风到燃烧室中液体燃料射流的雾化，再到天体物理体（如气态恒星）的运动，以及大气科学和气象学的基本预测。在这项工作中获得的分析性理解可能会对理解基本物理现象产生重要的影响，迄今为止，人们对这些现象知之甚少。除了基本的波浪运动和发生在水和空气之间的界面运动中的奇点之外，其他传统的例子包括空气和水之间的界面，云锋的运动，冰山的融化，两种压缩气体之间的基本不稳定性，激波中液体中的气泡的行为，以及通常通过首先雾化燃料喷射来增加表面积从而增加蒸发速率的液体燃料。本研究旨在分析气体、液体、等离子体中不断变化的不连续表面的运动，以及在经典相变模型的背景下。

Idealized fluid flows are modeled by the Euler equations of fluid dynamics; these are a coupled system on nonlinear conservation laws in (3+1)-dimensional space-time, and are the fundamental to all models of fluid motion. Compressible flows, in which the theory of sound is included, exhibit discontinuous wave profiles; examples include shock waves, contact discontinuities, moving material interfaces, moving vacuum boundaries, entropy waves, and a variety of additional discontinuous wave patterns. The analysis of such multi-dimensional wave patterns, and in particular, solutions of the Euler equations which propagate at least one surface of discontinuity, is fundamental to the understanding of basic physical phenomena. The goal of this research effort is to treat such multi-dimensional discontinuous wave profiles as moving free-boundary problems, and to develop a theory for the well-posedness of hyperbolic and degenerate hyperbolic systems, construct solutions which exhibit finite-time singularities wherein the propagating hypersurfaces collide with one another, study singular asymptotic limits such as vanishing viscosity limits and limits of zero surface tension, and develop a novel nonlinear stability theory for so-called hyperbolic-parabolic problems which are ubiquitous in models of phase transition.Multiphase fluid flows with moving interfaces play a central role in a multitude of physical and engineering applications, ranging from the creation of hurricanes due to wind blowing on top of the ocean surface to the atomization of liquid fuel jets in combustion chambers to the motion of astrophysical bodies such as gaseous stars, and to fundamental predictions in atmospheric science and meteorology. The analytical understanding gained in this work may have important ramifications in the understanding of basic physical phenomena, which is heretofore, poorly understood. In addition to basic wave motion and singularities that occurs in the motion of interfaces between water and air, other conventional examples include the interface between air and water, the motion of cloud fronts, the melting of ice-bergs, basic instabilities between two compressed gases, the behavior of a gas bubble in a liquid in a shock wave, and liquid fuels which are usually burned by first atomizing a fuel jet to increase the surface area and hence the evaporation rate. This proposed research aims to analyze the motion of evolving surfaces of discontinuity in gases, liquids, plasmas, as well as in the context of classical phase transition models.