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

小行星9622810 该项目的主要主题是I。非线性色散振荡， 二. Hele-Shaw流和III.斯托克斯流在项目I中， 目的是分析振荡的产生和传播 由散焦非线性薛定谔方程描述为色散 归零在微观上，振荡按照以下方式传播： Whitham方程所提出的方法是一个解决方案的建设 Whitham方程的一个快速变换。两个项目二 和III涉及的运动之间的界面更粘 流体和粘性较低的流体被认为是invicid。粘性 在第二个项目中，流体受达西定律支配， 最后一个是斯托克斯方程。重点分析了 表面张力下奇点的形成 斯托克斯流在3维中流动。所提出的方法将是 分析和计算。 %%% 在无碰撞激波中可以观察到非线性色散振荡 等离子体和光纤中的光学冲击。人着迷的 关于这个现象的是不同区域之间的相变。 我们的目标是了解从非振荡的相变 到振荡区域。Hele-Shaw流和Stokes流都是 多孔介质。众所周知，界面会不稳定 当粘性较小的流体驱动粘性较大的流体时。这种不稳定性 是威尔斯油井注水的主要原因， 对油藏工程的重要性。我们的目标是理解这一点 不稳定的细节。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。 ***

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. ***