Asymptotics of conservative PDE's and instability of vortex patches and filaments

保守偏微分方程的渐近性以及涡斑和细丝的不稳定性

• 批准号: 0510121
• 负责人: Daniel Spirn
• 金额: $ 3.17万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0510121&HistoricalAwards=false
• 关键词: Asymptotics; conservative; PDE; instability; vortex

Abstract, Proposal 0306398PI: Daniel P. SpirnTitle: Asymptotics of conservative PDEs and instability of vortex patches and filamentsABSTRACT: The proposed research concerns asymptotics of conservativepartial differential equations (PDE) and instability of the incompressible Euler equations in three specific areas. The first project concerns the long time behavior of vortex solutions in the nonlinear wave equation. Although recent progress has led to greater understanding of how concentrations behave in the nonlinear wave equation, very little is known about how such solutions radiate energy after long times; we will rigorously study this question. Second, we will study the nonlinear wave equation in asymptotically thin domains. Conventional wisdom holds that most PDE's with an asymptotically large aspect ratio will be well approximated simply by dropping the dimension of the PDE and relying only on the planar direction; this work will examine the precise timescales for which this practice is acceptable. The third project involves the nonlinear instability of various PDE's arising from the incompressible Euler equations, in which the governing dynamics reduce to the behavior of an interface -- such as in two-layer models and vortex patches.Often, physical phenomena are well modeled via nonlinear PDE's. Suchequations are exceedingly difficult to study, both numerically andtheoretically, yet their understanding is crucial to the further progress of many areas of physics and engineering. Two of the projects outlined above offer rigorous methods for simplifying classes of these equations, not only providing insight into the basic behavior of fluids and quantum field theory, but also greatly reducing the scope of numerical simulations. Such reductions in computational costs should aid physicists, computer scientists, and engineers. The third project endeavors to classify the stable structures in fluid dynamics, helping us to gain a better qualitative understanding of the way fluids behave.

摘要，提案0306398 PI：丹尼尔P. Spirn标题：渐近的保守偏微分方程和不稳定性的涡补丁和conventientsabstruct：拟议的研究涉及渐近的保守偏微分方程（PDE）和不稳定性的不可压缩欧拉方程在三个特定的领域。第一个项目是关于非线性波动方程涡解的长时间行为。虽然最近的进展使人们对浓度在非线性波动方程中的行为有了更深入的了解，但人们对这种溶液在长时间后如何辐射能量知之甚少;我们将严格研究这个问题。 其次，我们将研究渐近薄区域上的非线性波动方程。传统的智慧认为，大多数PDE的渐近大的纵横比将很好地近似简单地通过下降的PDE的尺寸，只依赖于平面方向，这项工作将检查精确的时间尺度，这种做法是可以接受的。第三个项目涉及各种PDE的非线性不稳定性，这些不稳定性来自不可压缩的Euler方程，在这些方程中，控制动力学归结为界面的行为--例如在两层模型和涡斑中。通常，物理现象可以通过非线性PDE很好地模拟。这样的方程是非常困难的研究，无论是数值和理论，但他们的理解是至关重要的许多领域的物理和工程的进一步发展。上面概述的两个项目提供了简化这些方程类的严格方法，不仅提供了对流体和量子场论基本行为的洞察，而且大大缩小了数值模拟的范围。这种计算成本的降低应该有助于物理学家，计算机科学家和工程师。第三个项目致力于对流体动力学中的稳定结构进行分类，帮助我们更好地定性了解流体的行为方式。