Mathematical Sciences: Nonlinear Wave Phenomena in Hydrodynamics

本文的研究工作涉及到当前应用数学领域的几个热点问题，特别是与连续介质力学中的非线性现象有关的问题。重点放在物理现象和数学理论用来描述非线性波动和稳定特性的流体力学。具体地说，将系统地研究非线性稳定性的一种新的理论方法。这一发展强调了平均流动/一次谐波相互作用的重要性，而不是更常见的涉及产生高次谐波的相互作用，作为剪切流动中三维扰动演变的驱动机制。已经考虑了一些特殊情况，相信这种办法可能对范围广泛的不稳定问题产生重要影响。PI打算考虑若干进一步的应用，以便充分评估其适用范围。

This research effort is concerned with several topics of current interest in applied mathematics, especially those associated with nonlinear phenomena in continuum mechanics. Emphasis is placed on both the physical phenomena and the mathematical theories used to describe nonlinear wave and stability characteristics in hydrodynamics. Specifically, a systematic study will be made of a new theoretical approach to nonlinear stability. This development emphasizes the importance of mean flow/first harmonic interactions, rather than the more usual ones that involve the generation of higher harmonics, as the driving mechanism underlying the evolution of 3-D disturbances in shear flows. Some special cases have already been considered and it is believed that this approach could have important consequences to a broad range of instability problems. The PI intends to consider a number of further applications so that the range of its applicability can be fully assessed.