Well-posedness of moving interface problems in perfect fluids

理想流体中运动界面问题的适定性研究建议摘要Steve ShkollerEuler方程无论有无表面张力，都被认为是大雷诺数下具有运动界面的多相流体流动的合适模型，即使在其他物理现象与流体运动耦合时，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，但对于无粘可压缩和不可压缩流体，这些移动边界PDE系统的局部适定性仍然是一个重大挑战。 这包括在真空中移动的单个流体质量以及由不连续表面分离的多相不混溶流体。最近，PI的这项拟议的研究工作已经开发出一种新的方法，适定性理论的流体中的移动界面问题，如三维不可压缩自由表面欧拉方程的边界上有或没有表面张力，耦合流体-结构相互作用问题。 其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规则性，并为其中存在的光滑的解决方案是可证明的。 该提案解决了可压缩气体在真空中运动的适定性，由自由边界可压缩欧拉方程建模，在边界处密度为零;涡面和不连续表面的适定性;以及相对论流体在真空中运动的适定性，由欧拉-爱因斯坦方程建模。 在分析中不进行无旋简化，以便考虑流体运动的全部范围。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，范围从由于风吹到海洋表面顶部而产生的飓风到燃烧室中液体燃料射流的雾化，如气态恒星等天体。从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发生的基本波运动和混合之外，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。

Well-posedness of moving Interface Problems in Perfect FluidsAbstract of Proposed ResearchSteve ShkollerThe Euler equations with, or without, surface tension are recognized as a suitable model for multiphase fluids flows with moving interfaces at large Reynolds number, and 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 local well-posedness of these systems of moving boundary PDE for inviscid compressible and incompressible fluids remains a significant challenge. This includes a single mass of fluid moving in vacuum as well as multi-phase immiscible fluids separated by surfaces of discontinuity. Recently, the PI of this proposed research effort has developed a novel approach to well-posedness theories for moving interface problems in fluids such as the 3D incompressible free-surface Euler equations with or without surface tension on the boundary, and coupled fluid-structure interaction problems. The fundamental idea relies on new anisotropic smoothing operators which 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. The proposal addresses the well-posedness of the motion of a compressible gas in vacuum, modeled by the free-boundary compressible Euler equations with density vanishing at the boundary; well-posedness of vortex sheets and surfaces of discontinuity; and well-posedness for the motion of a relativistic fluid in vacuum, modeled by the Euler-Einstein equations. No irrotationality simplifications will be made in the analysis so that the full range of fluid motion can be considered.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 from 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 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 Richtmyer-Meshkov instabilities between two 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. 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.