Analysis of moving interface problems in fluid dynamics

理想化的流体流动由流体动力学的欧拉方程建模;这些方程是（3+1）维时空中非线性守恒律的耦合系统，并且是所有流体运动模型的基础。可压缩流，其中包括声音的理论，表现出不连续的波形;例子包括冲击波，接触不连续，移动的材料界面，移动的真空边界，熵波，和各种其他不连续的波图案。这种多维波型的分析，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。 本研究工作的目标是将这种多维不连续波剖面视为移动自由边界问题，并发展双曲和退化双曲系统的适定性理论，构造表现出有限时间奇异性的解，其中传播超曲面相互碰撞，研究奇异渐近极限，如零粘度极限和零表面张力极限，并发展了一种新的非线性稳定性理论，用于相变模型中普遍存在的所谓双曲-抛物问题。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，范围从由于风吹在海面上而产生的飓风到燃烧室中液体燃料射流的雾化，天体物理学的机构，如气态恒星，并在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对理解基本物理现象产生重要影响，而迄今为止，对基本物理现象的理解很少。除了在水和空气之间的界面的运动中出现的基本波动和奇异性之外，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、两种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、以及通常通过首先雾化燃料射流以增加表面积并因此增加蒸发速率来燃烧的液体燃料。 本研究旨在分析气体、液体、等离子体以及经典相变模型中不连续表面的运动。

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.