Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

本项目将继续对完美流体力学中出现的非线性椭圆问题进行数学研究。重点将放在波在分层介质中传播的分析研究上。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旋环的存在。例如，由于盐度或温度的变化，密度分层产生了内波。实验工作显示了内波现象的惊人例子，海洋观测揭示了振幅高达100米的内波。这项工作将集中于分析两流体系统中的波，其中两种密度恒定的流体沿着光滑的密度剖面相遇。在这种情况下，中心流形理论可以用来研究用常微分方程代替非线性偏微分方程的波动问题。近年来应用全局分析方法研究具有Heaviside函数分布的涡环已被证明是有效的。工作将继续努力去理解描述涡旋位移函数的方程解的水平集的性质。该方程具有跳跃不连续，有时可以确定其存在性，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.