Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

该项目将继续对完全流体流体力学中出现的非线性椭圆型问题进行数学研究。重点将放在分层介质中波传播的分析研究上。来自非线性分析和偏微分方程的技术构成了这些研究的基础。主要目标是更好地了解内波的性质和涡环的存在。例如，内波是由盐度或温度变化引起的密度分层引起的。实验工作展示了内波现象的显著例子，海洋观测显示内波的幅度高达100米。这项工作将集中在双流体系统中的波的分析，其中两个恒定密度的流体沿着平滑的密度剖面相遇。在这种情况下，可以用中心流形理论来研究常微分方程组，取代波动运动的非线性偏微分方程组。最近，整体分析在研究具有Heaviside函数分布的涡环方面的应用已被证明是有效的。将继续努力了解描述涡流位移函数的方程的解的水平集的性质。这个方程有跳跃间断，有时可以确定它的存在，但水平集的性质仍然是一个谜。

Work to be done on this project continues mathematical research on nonlinear elliptic problems arising in perfect-fluid hydrodynamics. Emphasis will be placed on the analytical study of the propagation of waves in stratified media. Techniques from nonlinear analysis and partial differential equations form the basis for these studies. The primary goals are to understand better the nature of internal waves and the presence of vortex rings. The internal waves arise from density stratification due to changes in salinity or temperature, for example. Experimental work shows striking examples of internal wave phenomena, and oceanic observations reveal internal waves up to 100 meters in amplitude. This work will concentrate on the analysis of waves in the two-fluid system where two fluids of constant density meet along a smooth density profile. In this case, center-manifold theory can be applied to study the ordinary differential equations which replace the nonlinear partial differential equations for the wave motion. Recent applications of global analysis to study vortex rings which have a Heaviside function distribution have proved effective. Work will continue in an effort to understand the properties of the level set of the solutions to the equation describing the vortex displacement function. The equation has jump discontinuities, for which existence can sometimes be established, but the nature of the level sets remains a mystery.