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

摘要：本文研究了守恒偏微分方程（PDE）的渐近性和不可压缩欧拉方程在三个特定区域的不稳定性。第一个项目涉及非线性波动方程中涡旋解的长时间行为。尽管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，但人们对这些溶液在长时间后如何辐射能量知之甚少；我们将认真研究这个问题。其次，我们将研究渐近薄域上的非线性波动方程。传统观点认为，大多数具有渐近大宽高比的偏微分方程可以简单地通过降低偏微分方程的尺寸并仅依赖于平面方向来很好地近似；这项工作将检查这种做法是可以接受的精确时间尺度。第三个项目涉及由不可压缩欧拉方程引起的各种PDE的非线性不稳定性，其中控制动力学减少到界面的行为-例如在两层模型和漩涡斑块中。通常，物理现象可以通过非线性偏微分方程很好地建模。这样的方程在数值上和理论上都很难研究，但对它们的理解对物理学和工程学许多领域的进一步发展至关重要。上面列出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见解，而且大大减少了数值模拟的范围。计算成本的这种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理解流体的行为方式。

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.