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

9622810项目的主要主题是：I.非线性色散振荡，II.Hele-Shaw流和III.Stokes流。在项目I中，主要目的是分析当色散趋于零时，由散焦非线性薛定谔方程描述的振荡的产生和传播。在微观上，振荡按照Whitham方程传播。所提出的方法是通过速度图变换来构造Whitham方程的解。方案二和方案三都涉及粘性较强的流体和粘性较低的流体之间的界面运动，被认为是非粘性的。在第二个方案中，粘性流体遵循达西定律，在最后一个方案中，粘性流体遵循斯托克斯方程。重点讨论了三维表面张力、Hele-Shaw和Stokes流的奇点形成类型。所提出的方法将是解析和计算的。在等离子体中的无碰撞冲击和光纤中的光学冲击中都能观察到%%-%%的非线性色散振荡。这种现象令人着迷的是不同地区之间的相变。我们的目标是了解从非振荡区到振荡区的相变。Hele-Shaw流动和Stokes流动都是多孔介质中的流动。众所周知，当粘度较低的流体驱动粘度较大的流体时，界面将不稳定。这种不稳定性是导致油井水驱的原因，对油藏工程具有明显的重要性。其目的是详细了解这种不稳定性。首席研究员建议使用一些新的解析和数值方法来研究这些问题。***

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. ***