Well-posedness of moving interface problems in perfect fluids

欧拉方程被认为是一个合适的模型，多相流体流动的移动边界和界面在大雷诺数，他们作为基本的数学模型，即使当其他物理现象耦合到流体运动。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，但这些双曲移动自由边界PDE系统的存在性理论仍然是一个重大的挑战。这包括单个流体、气体或液体的质量，在真空内部移动，在流动中产生简并，以及由不连续表面分离的多相不混溶流体，速度分量在不连续表面上经历跳跃。最近，该项目的主要研究者一直在开发一套新的分析工具，旨在建立多维移动自由边界双曲问题的存在性理论和适定性定理，其中自由表面的几何形状与领先阶流体的运动相互作用。这些分析工具适用于三维不可压缩和可压缩的自由表面欧拉方程的边界上有或没有表面张力，耦合的流体-结构相互作用的问题。 的基本思想依赖于新的各向异性平滑算子，允许近似的欧拉方程，保留几何结构的运输和边界的正则性，并存在光滑的解决方案是可证明的，一类新的退化抛物近似的特征和退化双曲守恒律系统。 该提案解决了多维可压缩气体在所谓的物理真空奇点中运动的适定性问题，该奇点由自由边界可压缩欧拉方程模拟，声速在边界处以到真空距离的平方根的速率消失;超音速二维涡面和不连续表面的适定性;真空中相对论性流体运动的适定性和适定性。由欧拉方程模拟的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洋表面上而产生的飓风到燃烧室中液体燃料喷射的雾化，再到诸如气态恒星之类的天体物理体的运动。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，迄今为止对这些现象的理解很少。除了在水和空气之间的界面的运动中发生的基本波动和混合之外，其他常规示例包括在冲击波相互作用下的空气和氦气之间的界面、两种气体之间的所谓的Richtmyer-Meshkov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。我们还可以添加喷雾行为的预测，其中初始雾化是喷雾的最关键和最不了解的方面。 理解瑞利-泰勒不稳定性中发生的短时非线性平衡对于理解射流应该是非常重要的，当毛细效应由于波的长度大于直径而变得很大时，射流变得不稳定，从而分裂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.