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

摘要，建议0306398PI：Daniel P.SpirnTitle：守恒偏微分方程组的渐近性和涡片和涡丝的不稳定性摘要：拟议的研究涉及三个特定区域内守恒偏微分方程组的渐近性和不可压缩欧拉方程的不稳定性。第一个项目是关于非线性波动方程中涡旋解的长时间行为。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多的了解，但对于这种解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。传统的观点认为，大多数具有渐近大长宽比的偏微分方程只需降低偏微分方程的维度并仅依赖于平面方向即可很好地逼近；这项工作将检查这种做法可以接受的精确时间标度。第三个项目涉及由不可压缩欧拉方程产生的各种偏微分方程组的非线性不稳定性，其中控制动力学归结为界面的行为--例如在两层模型和涡流片中。通常，物理现象通过非线性偏微分方程组来很好地模拟。成功的方程无论在数值上和理论上都是非常困难的，但它们的理解对于许多物理和工程领域的进一步发展是至关重要的。上面概述的两个项目提供了简化这些方程类别的严格方法，不仅提供了对流体和量子场理论的基本行为的洞察，还极大地缩小了数值模拟的范围。这种计算成本的降低应该会对物理学家、计算机科学家和工程师有所帮助。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地了解流体的行为方式。

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.