Mathematical Sciences: Mathematical Problems From Nonlinear Dispersive Oscillations, Hele-Shaw and Stokes Flows

数学科学：非线性色散振荡、赫勒肖和斯托克斯流的数学问题

• 批准号: 9622810
• 负责人: Fei-Ran Tian
• 金额: $ 5.69万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1999-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622810&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Nonlinear; Dispersive

9622810 Tian The main themes on the project are I. Nonlinear Dispersive Oscillations, II. Hele-Shaw Flows and III. Stokes Flows. In project I, the main purpose is to analyze the generation and propagation of oscillations described by the defocusing nonlinear Schrodinger equation as dispersion goes to zero. Microscopically, the oscillations propagate according to the Whitham equations. The proposed method is a construction of solutions of the Whitham equations via a hodograph transform. Both projects II and III concern the motion of an interface between a more viscous fluid and a less viscous fluid to be regarded as invicid. The viscous fluid is governed by Darcy's law in the second project, and by the Stokes equations in the last one. The emphases are on the types of singularity formation in the presence of surface tension, Hele-Shaw and Stokes flows in dimension 3. The proposed methods will be both analytical and computational. %%% Nonlinear dispersive oscillations can be observed in collisionless shocks in plasmas and optical shocks in optical fibers. What is fascinating about this phenomenon is the phase transition between different regions. The goal is to understand the phase transition from non-oscillatory to oscillatory regions. Both Hele-Shaw and Stokes flows are flows in porous media. It is well known that the interface will be unstable when a less viscous fluid drives a more viscous fluid. This instability is responsible for water flooding of oil wells, and is of obvious importance to oil reservoir engineering. The aim is to understand this instability in detail. The principal investigator proposes to use some new analytical and numerical techniques to study these problems. ***

[9622810]田：本课题的主要研究主题是：1 .非线性色散振荡；Hele-Shaw流和3。斯托克斯流。在项目一中，主要目的是分析当色散趋于零时，由离焦非线性薛定谔方程描述的振荡的产生和传播。微观上，振荡根据惠瑟姆方程传播。所提出的方法是通过一个全息变换来构造Whitham方程的解。项目二和项目三都涉及黏性较强的流体和黏性较弱的流体之间的界面运动，被认为是无黏性的。粘性流体在第二个项目中由达西定律控制，在最后一个项目中由斯托克斯方程控制。重点讨论了在存在表面张力、Hele-Shaw流和Stokes流的情况下奇点形成的类型。所提出的方法将是分析和计算的。在等离子体中的无碰撞冲击和光纤中的光学冲击中可以观察到非线性色散振荡。这个现象的奇妙之处在于不同区域之间的相变。我们的目标是了解从非振荡区到振荡区的相变。Hele-Shaw流和Stokes流都是多孔介质中的流动。众所周知，当粘性较小的流体驱动粘性较大的流体时，界面将不稳定。这种不稳定性是造成油井水驱的主要原因，对油藏工程具有重要意义。目的是详细了解这种不稳定性。主要研究者建议使用一些新的分析和数值技术来研究这些问题。＊＊＊