Long-time Behavior of Some Dispersive and Fluid Equations

本项目致力于研究数学物理和流体动力学中出现的几个偏微分方程。我们试图结合不同数学学科的不同思想，在这些问题上取得进展。第一个问题涉及到在一个弯曲的底部地形存在下的自由水面的运动。作为理解海洋中波浪复杂动力学的第一步，我们将考虑平坦水面的小扰动，并分析这种扰动如何在长时间内传播。第二个问题是关于与薛定谔有关的长期现象？S方程在周期框中。我们打算证明盒子的形状和大小的精细数论性质将如何影响薛定谔波的长时间行为。第三个问题涉及磁流体动力学（MHD）方程，它描述了许多重要的等离子体系统，如太阳。特别地，在强背景磁场的存在下，我们将研究在消失粘度极限下阿尔芬波的形成。第四个也是最后一个问题与流体动力学中的无粘阻尼现象有关，它是不稳定的，需要对流体施加一些微妙的条件。我们扩展了之前在这个主题上的工作，并打算研究在违反这些条件的情况下，无粘阻尼何时以及如何失效。这四个问题中的每一个都有特定的困难，需要一种新的方法。在第一个问题中，所涉及的方程是一个变系数的非线性色散方程，因此现有的已成功处理常系数色散方程的方法不再适用。然而，我们计划将这种基于傅里叶空间分析的方法与适当的物理空间方法相结合来研究这个问题。在第二个问题中，我们考虑薛定谔？缺乏张量积结构的非矩形环面上的S方程。出于这个原因，我们将引入数论和谐波分析中更先进的技术，例如解耦方法和丢芬图系统解的计数公式。第三个问题涉及一个弱耗散系统，该系统在消失的粘度极限下变得色散，因此我们需要将适用于色散方程的方法扩展到耗散环境，在傅里叶方面相当于将耗散衰减与固定相结合。在第四个问题中，主要挑战是将我们之前工作中的线性分析扩展到无限时间，这需要傅里叶空间和物理空间技术的结合（例如，在傅里叶空间和物理空间中使用指数权重进行能量估计）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is devoted to the study of several partial differential equations that arise in mathematical physics and fluid dynamics. We attempt to make progress on these problems by combining different ideas from different subjects of mathematics. The first problem concerns the motion of the free water surface in the presence of a curved bottom topography. As a first step towards understanding the complex dynamics of waves in the ocean, we will consider small perturbations to the flat water surface and analyze how such perturbations propagate over long time periods. The second problem is about the long-time phenomenon associated with the Schrodinger?s equation in a periodic box. We intend to demonstrate how the fine number theoretic properties of the shape and the size of the box will affect the long-time behavior of Schrodinger waves. The third problem involves the magnetohydrodynamics (MHD) equations, which describes many important plasma systems such as the sun. In particular, in the presence of a strong background magnetic field, we will study the formation of Alfven waves in the vanishing viscosity limit. The fourth and last problem is related to the phenomenon of inviscid damping in fluid dynamics, which is unstable and requires some subtle conditions on the fluid. We extend our previous work on this topic, and intend to study when and how inviscid damping may fail, provided that these conditions are violated.There are specific difficulties in each of the four problems that require a novel approach. In the first problem, the equation involved is a nonlinear dispersive equation with variable coefficients, so the existing methods which have been successful in treating constant coefficient dispersive equations are no longer applicable. Nevertheless, we plan to combine this method, which is based on Fourier space analysis, with appropriate physical space methods to study this problem. In the second problem we are considering Schrodinger?s equation on non-rectangular tori that lack tensor product structure. For this reason we will bring in more advanced techniques from number theory and harmonic analysis, such as the decoupling methods, and counting formulas for solutions to Diophantine systems. The third problem involves a weakly dissipative system which in the vanishing viscosity limit becomes dispersive, and thus we will need to extend the methods that are suitable for dispersive equations to the dissipative context, which on the Fourier side amounts to combining dissipation decay with stationary phase. In the fourth problem, the main challenge is extending the linear analysis in our previous work to infinite time, which requires a combination of Fourier space and physical space techniques (for example, energy estimates with exponential weights both in Fourier and in physical space).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.