Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

数学科学：流体动力学的非线性偏微分方程

• 批准号: 9000110
• 负责人: Robert E Turner
• 金额: $ 7.1万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-04-15 至 1993-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9000110&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

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.

该项目要做的工作继续对完美流体流体动力学中出现的非线性椭圆问题进行数学研究。 重点将放在分层介质中波传播的分析研究上。 非线性分析和偏微分方程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，由于盐度或温度的变化，密度分层会产生内波。 实验工作展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双流体系统中的波，其中两种密度恒定的流体沿着平滑的密度剖面相遇。 在这种情况下，可以应用中心流形理论来研究常微分方程，以代替波动运动的非线性偏微分方程。 最近应用全局分析来研究具有亥维赛函数分布的涡环已被证明是有效的。 我们将继续努力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性，有时可以建立跳跃不连续性，但水平集的性质仍然是个谜。