Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

在这个项目上要做的工作继续数学 理想流体中非线性椭圆问题的研究 流体力学 重点将放在分析研究上 波在分层介质中的传播。 技术从 非线性分析和偏微分方程的形式， 这些研究的基础。 主要目标是了解 更好的内波的性质和涡流的存在 环. 内波产生于密度分层， 例如盐度或温度的变化。 实验 工作显示了内部波现象的惊人例子， 海洋观测显示， 振幅 这项工作将集中在波的分析 在两种恒定密度的流体相遇的双流体系统中 沿着平滑的密度分布。 在这种情况下，中心歧管 理论可以用来研究常微分方程 方程取代非线性偏微分方程 波动方程 整体分析在涡环研究中的应用 有一个Heaviside函数分布， 有效 将继续努力了解 方程解的水平集的性质 描述涡流位移函数。 该方程有 跳跃不连续性，其存在有时可能是 建立，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.