Well-posedness of moving interface problems in perfect fluids

欧拉方程被认为是大雷诺数下具有移动边界和界面的多相流体流动的合适模型，即使在其他物理现象耦合到流体运动中，欧拉方程也可以作为基本的数学模型。尽管对这些复杂的非线性方程进行了两个多世纪的数学分析，但这些双曲运动自由边界偏微分方程系统的存在性理论仍然是一个重大的挑战，它可以模拟理想的可压缩和不可压缩流体的流动。这包括在真空中移动的单一质量的流体、气体或液体，在流动中产生简并，以及由不连续表面分离的多相不混相流体，在该表面上速度分量经历跳跃。最近，该项目的首席研究员开发了一套新的分析工具，旨在建立多维运动自由边界双曲问题的存在性理论和适定性定理，其中自由曲面的几何形状与流体的运动相互作用。这些分析工具适用于边界上有无表面张力的三维不可压缩和可压缩自由表面欧拉方程，以及耦合流固耦合问题。基本思想依赖于新的各向异性平滑算子，它允许欧拉方程的近似保留了传输和边界规则的几何结构，并且光滑解的存在性是可证明的，以及一类新的退化抛物近似的特征和退化双曲系统的守恒定律。该提议解决了所谓的物理真空奇点中多维可压缩气体运动的适定性，该运动由自由边界可压缩欧拉方程模拟，声速在边界以到真空距离的平方根的速率消失；超声速二维涡片和不连续面的适位性；以及由欧拉-爱因斯坦方程模拟的真空中相对论性流体运动的适定性。由欧拉方程模拟的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。在这项工作中获得的分析性理解可能会对理解迄今为止知之甚少的基本物理现象产生重要影响。除了在水和空气之间的界面运动中发生的基本波动和混合之外，其他传统的例子包括激波相互作用下空气和氦之间的界面，所谓的两种气体之间的richmyer - meshkov不稳定性，激波中液体中的气泡行为，以及通常通过首先雾化燃料射流来增加表面积从而增加蒸发速率的液体燃料。我们还可以加入喷雾行为的预测，其中初始雾化既是喷雾中最关键的也是最不容易理解的方面。理解在瑞利-泰勒不稳定性中发生的短期非线性平衡对于理解射流是非常重要的，当毛细效应由于波的长度大于直径而变得不稳定，从而分解成相对较大的液滴流。

The Euler equations are recognized as a suitable model for multiphase fluid flows with moving boundaries and interfaces at large Reynolds number, and they serve as the basic mathematical model even when other physical phenomena are coupled to the fluid motion. Despite more than two centuries of mathematical analysis of these complicated nonlinear equations, the existence theory for these systems of hyperbolic moving free-boundary PDE, which model ideal compressible and incompressible fluid flow, remains a significant challenge. This includes a single mass of fluid, gas, or liquid, moving inside of a vacuum that creates degeneracy in the flow, as well as multiphase immiscible fluids separated by surfaces of discontinuity, across which velocity components experience jumps. Recently, the Principal Investigator of this project has been developing a novel set of analytical tools designed to establish existence theories and well-posedness theorems for multidimensional moving free-boundary hyperbolic problems, wherein the geometry of the free-surface interacts with the motion of the fluid at leading order. These analytical tools apply to the 3-dimensional incompressible and compressible free-surface Euler equations with or without surface tension on the boundary, and coupled fluid-structure interaction problems. The fundamental ideas rely on new anisotropic smoothing operators that permit approximations of the Euler equations that retain the geometric structures of transport and boundary regularity, and for which existence of smooth solutions is provable, and a new class of degenerate parabolic approximations to characteristic and degenerate hyperbolic systems of conservation laws. The proposal addresses the well-posedness of the motion of a multidimensional compressible gas in the so-called physical vacuum singularity, modeled by the free-boundary compressible Euler equations with sound speed vanishing at the boundary at the rate of the square-root of the distance to vacuum; well-posedness of supersonic 2-D vortex sheets and surfaces of discontinuity; and well-posedness for the motion of a relativistic fluid in vacuum, modeled by the Euler-Einstein equations.Multiphase fluid flows with moving interfaces, modeled by the Euler equations, 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. The analytical understanding gained in this work may have important ramifications in the understanding of basic physical phenomena, which has hitherto been poorly understood. In addition to basic wave motion and mixing that occurs in the motion of interfaces between water and air, other conventional examples include the interface between air and helium under shock wave interaction, the so-called Richtmyer-Meshkov instabilities between two gases, the behavior of a gas bubble in a liquid in a shock wave, and liquid fuels that are usually burned by first atomizing a fuel jet to increase the surface area and hence the evaporation rate. We can also add the prediction of spray behavior, for which the initial atomization is both the most critical and the least understood aspect of the spray. Understanding the short-time nonlinear balance that occurs in the Rayleigh-Taylor instability should be quite important for the understanding of jets, which become unstable when capillary effects are large due to waves longer than the diameter, thus breaking up into a stream of relatively large drops.