The Interplay of Evolution Partial Differential Equations and Their Model Equations

演化偏微分方程及其模型方程的相互作用

• 批准号: 1612931
• 负责人: Nathan Totz
• 金额: $ 10.21万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612931&HistoricalAwards=false
• 关键词: Interplay; Evolution; Partial; Differential; Equations

In mathematical physics, equations that predict the behavior of many important physical systems (including water waves, signals in fiber optic cables, and quantum particles) have disparate scales that make the problem of understanding their solutions extremely challenging, be it through analysis or through computer simulations. However, the problem can be simplified by focusing on one salient aspect of the phenomenon and ignoring the others. The "model equations" that result from this process are generally easier to handle than the original equation. However, model equations are often derived using heuristic arguments, and attempts to confirm the accuracy of the models numerically can be inconclusive. One aim of this research project is to mathematically justify that model equations accurately describe the full equation. In some cases, analysis of solutions to such simplified model equations remains difficult, even to the point where basic facts about the behavior of such model equations (such as whether they can be solved for all times) are not known. A good theoretical understanding of these fundamental properties is important for providing confidence that the model in question is accurate. This project will explore ways in which an equation interacts with its model equations and vice versa, with the goals of a better understanding of both problems. This project deals with research involving partial differential equations (PDEs), specifically the relationship between evolution PDEs and model equations derived from them by asymptotic expansion methods. The first and most fundamental question of study from a mathematical standpoint is rigorous justification: to give a mathematical proof that a model equation derived from a formal argument provides a good approximation to the original PDE in the appropriate regime. This entails providing quantitative bounds on the errors incurred in approximation. Since such models often involve small parameters, one must have a good understanding of the well-posedness of the underlying problem for large initial data and over long time scales, both of which involve technical challenges. The other aspect of this project is to explore the idea of constructing, for a given equation known to arise from such a modeling procedure, an artificial equation constructed to force existence of the model equation for all time. This provides a new method for demonstrating global existence of solutions to equations which may be lacking structure for applying existing techniques, such as a coercive conserved energy. The first step is to demonstrate that this method applies in the context of nonlinear Schrodinger equations, which is one of the fundamental equations of mathematical physics. This project aims also to generalize the method to analyze other important model equations in which conserved quantities are either absent or otherwise unsuitable for use. A famous example of such an obstacle is the "super-criticality barrier" that prevents the establishment of global existence in a wide variety of problems.

在数学物理中，预测许多重要物理系统(包括水波、光缆中的信号和量子粒子)行为的方程具有不同的尺度，这使得通过分析或计算机模拟来理解它们的解决方案的问题具有极大的挑战性。然而，通过专注于现象的一个显著方面而忽略其他方面，问题可以被简化。这一过程产生的“模型方程”通常比原始方程更容易处理。然而，模型方程通常是使用启发式论证得出的，试图从数字上确认模型的准确性可能是不确定的。这个研究项目的目的之一是从数学上证明模型方程准确地描述了完整的方程。在某些情况下，对这种简化的模型方程的解的分析仍然很困难，甚至到了关于这种模型方程的行为的基本事实(例如它们是否可以在任何时候都能解)都不知道的地步。对这些基本性质的良好理论理解对于确保所讨论的模型是准确的是很重要的。这个项目将探索方程与其模型方程相互作用的方式，目的是更好地理解这两个问题。本课题主要研究偏微分方程组，特别是发展偏微分方程组与用渐近展开方法得到的模型方程之间的关系。从数学角度研究的第一个也是最基本的问题是严格的论证：给出一个数学证明，证明从形式论证得出的模型方程在适当的制度下提供了对原始偏微分方程组的良好近似。这需要为近似中产生的误差提供量化的界限。由于这类模型往往涉及较小的参数，人们必须很好地理解潜在问题对于大初始数据和长时间尺度的适定性，这两者都涉及技术挑战。这个项目的另一个方面是探索这样一个想法，即对于已知从这种建模过程中产生的给定方程，构造一个人造方程，以迫使该模型方程始终存在。这为证明方程整体解的存在性提供了一种新的方法，这些方程可能缺乏应用现有技术的结构，例如强制守恒能量。第一步是证明这种方法适用于非线性薛定谔方程，这是数学物理的基本方程之一。该项目还旨在推广这种方法来分析其他重要的模型方程，在这些方程中要么没有守恒量，要么不适合使用。这种障碍的一个著名例子是“超临界障碍”，它阻止在各种各样的问题中建立全球存在。