Analysis of moving interface problems in fluid dynamics

理想的流体流动由流体动力学的欧拉方程来模拟；这些方程是(31)维时空中的非线性守恒律的耦合系统，是所有流体运动模型的基础。包括声音理论在内的可压缩流动呈现出不连续的波型，例如激波、接触不连续、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的波型。分析这种多维波型，特别是传播至少一个不连续表面的欧拉方程的解，是理解基本物理现象的基础。这项研究的目的是将这种多维不连续波视为移动的自由边界问题，发展双曲和退化双曲组的适定性理论，构造传播超曲面相互碰撞的有限时间奇异性解，研究奇异渐近极限，如零粘性极限和零表面张力极限，并发展一种新的非线性稳定性理论来解决在相变模型中普遍存在的所谓双曲-抛物型问题。具有运动界面的多相流体流动在许多物理和工程应用中起着核心作用，从海洋表面吹起的风造成飓风，到燃烧室中液体燃料射流的雾化，到气态恒星等天体的运动，再到大气科学和气象学的基本预测，不一而足。在这项工作中获得的分析性理解可能会对理解基本的物理现象产生重要的影响，而到目前为止，人们对基本物理现象的理解还很少。除了水和空气之间的界面运动中出现的基本波动和奇点之外，其他常规的例子包括空气和水之间的界面、云锋的运动、冰山的融化、两个压缩气体之间的基本不稳定性、冲击波中液体中气泡的行为，以及液体燃料，通常通过首先雾化燃料喷射器来增加表面积从而增加蒸发速度来燃烧。这项拟议的研究旨在分析气体、液体、等离子体中不连续面的运动，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.