New asymptotic methods and applications to physical problems

该项目的重点是进一步发展在一些重要的物理应用中出现的线性和非线性偏微分方程式的构造性和严格的渐近分析。这些应用包括：(A)叠加在通道或管道中振荡流动上的小扰动的演化，(B)通道和管道流动中自持状态的存在和稳定性，被认为是中间雷诺数湍流的巨型极限；(C)Hele-Shaw流动中稳定平移的手指和气泡的非线性稳定性，对于小的非零表面张力，感兴趣的图样形成；(D)氢原子在任意频率和幅度的时间周期偶极子场中的电离。我们对流体运动和量子力学中所产生的方程的解的理解仍然非常有限。目前的项目开发了迫切需要的新方法来研究这些问题。在数学之外的直接应用中，我们注意到，了解湍流流动可以导致流体通过管道进行更节能的输送。此外，更好地理解电离过程也有许多应用，包括非常强的激光在大气中的最佳传输。该项目支持的对学生的培训将有助于为下一代在美国保持高质量的研究。

The project focuses on further development of constructive andrigorous asymptotic analysis of linear and nonlinear partialdifferential equations that arise in a number of important physicalapplications. The applications include (a) evolution of smalldisturbances superposed on an oscillatory flow in a channel or pipe,of interest in transition to turbulence, (b) existence and stabilityof self-sustaining states in channel and pipe flows, believed to beomega-limits in intermediate Reynolds number turbulent flow, (c)nonlinear stability of steadily translating fingers and bubbles inHele-Shaw flow for small nonzero surface tension, of interest inpattern formation, (d) ionization of Hydrogen atoms in time-periodicdipole fields of arbitrary frequency and amplitude. Our understanding of the solutions of equations arising in fluidmotion and quantum mechanics is still very limited. The presentproject develops much needed new methods to study these problems.Among the immediate applications outside mathematics we note forinstance that understanding turbulent fluid flow can lead to a moreenergy-efficient transportation of fluids through pipes. Also, betterunderstanding of ionization processes has many applications, includingoptimal propagation of very intense laser beams through theatmosphere. Training of students --supported in the project-- willhelp maintain the high quality of research in the United States forthe coming generation.