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摘要steve shkolller有或没有表面张力的欧拉方程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，但对于无粘可压缩和不可压缩流体，这些运动边界偏微分方程系统的局部适定性仍然是一个重大挑战。这包括在真空中运动的单一质量流体以及由不连续表面分离的多相不混相流体。最近，该研究项目的PI已经开发了一种新的方法来解决流体中移动界面问题的适定性理论，例如边界上有或没有表面张力的三维不可压缩自由欧拉方程，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以证明的。该方案解决了真空中可压缩气体运动的适定性，用密度在边界处消失的自由边界可压缩欧拉方程来建模；旋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。为了考虑流体运动的全范围，在分析中不作无旋转性的简化。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。从这项工作中获得的分析理解可能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，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。

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.