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Generic Singularities and Fine Regularity Structure for Nonlinear Partial Differential Equations

非线性偏微分方程的一般奇异性和精细正则结构

基本信息

批准号：
2154201
负责人：
Tien Khai Nguyen
金额：
$ 13.72万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154201&HistoricalAwards=false
关键词：
Generic Singularities Fine Regularity Structure

项目摘要

Numerical solutions of nonlinear partial differential equations (PDE) play a crucial role in a variety of applications. However, if the solutions do not have sufficient regularity, the performance of numerical algorithms and their computational accuracy is a challenging issue. In this project, the PI will study some fundamental questions regarding the regularity of solutions for various classes of PDE modeling nonlinear waves, also in the presence of nonlocal source terms. A major focus of the research will be on the emergence of singularities, such as shock waves, for generic solutions. The expected results will provide an accurate asymptotic description of how new singularities are formed, and how they interact with each other, valid for almost all initial data. This will lead to a new class of numerical schemes, with high-order accuracy and a wide range of applications. In addition, the project will provide a training ground for both undergraduate and graduate students.This research project contains two main parts. The first part is concerned with the fine regularity structure of solutions to Hamilton-Jacobi equations. Here, the goals are to (i) deepen the analysis of the metric entropy of sets of solutions to Hamilton-Jacobi equations, study SBV regularity, and develop new techniques that cover a wider class of equations; and (ii) investigate the fine properties of generalized monotone functions, study the propagation of singularities, and establish new regularity estimates and controllability results for Hamilton Jacobi equations. The second part of the project will focus on generic singularities for nonlinear balance laws, also in the presence of nonlocal terms. For a wide class of such equations, it is well known that solutions with smooth initial data can lose regularity in finite time. However, they can be extended in a weak sense beyond the time when the first derivatives blowup. A major difficulty in understanding entropy weak solutions is that nonlocal source terms have a huge influence on singularity formation. For this reason, many interesting properties of entropy weak solutions, such as uniqueness, shock formation, shock interactions, and the structure of singularities, are still far from being well understood. In this part of the project the PI aims to (i) develop a quantitative analysis of the number of shocks for entropy weak solutions and study their generic regularity for various models of nonlinear waves and for nonlinear hyperbolic systems of conservation laws; and (ii) provide a detailed description of shock formation and wave breaking of entropy weak solutions for nonlocal balance laws as well as trace their impact on BV regularity and stability results. The approach that will be pursued, taking advantage of the piecewise regularity of solutions, seeks to reduce an equation defined on a space with low regularity to an equation on a more regular space coupled with an ODE on a finite dimensional manifold. In cases where the solutions of interest are known to be generically piecewise smooth, this can have an impact in the broader field of numerical analysis, suggesting new high-order computational algorithms.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.

非线性偏微分方程（PDE）的数值解在许多应用中起着至关重要的作用。然而，如果解不具有足够的规律性，则数值算法的性能和计算精度是一个具有挑战性的问题。在这个项目中，PI将研究一些关于非线性波的各种PDE模型解的正则性的基本问题，也包括非局部源项的存在。研究的一个主要焦点将是奇点的出现，如冲击波，对于通用的解决方案。预期的结果将提供一个准确的渐近描述如何形成新的奇点，以及它们如何相互作用，对几乎所有的初始数据有效。这将导致一类新的数值格式，具有高阶精度和广泛的应用。此外，该项目将为本科生和研究生提供一个训练场地。本研究项目包括两个主要部分。第一部分讨论了Hamilton-Jacobi方程解的精细正则结构。在这里，目标是(i)深化对Hamilton-Jacobi方程解集的度量熵的分析，研究SBV规则性，并开发涵盖更广泛类型方程的新技术；（ii）研究广义单调函数的精细性质，研究奇异性的传播，建立Hamilton Jacobi方程的新的正则性估计和可控性结果。该项目的第二部分将集中于非线性平衡定律的一般奇点，也是在非局部项的存在下。对于一类广泛的此类方程，众所周知，具有光滑初始数据的解在有限时间内会失去正则性。然而，在一阶导数崩溃之后，它们可以在弱意义上得到扩展。理解熵弱解的一个主要困难是非局域源项对奇点的形成有很大的影响。由于这个原因，熵弱解的许多有趣的性质，如唯一性、激波形成、激波相互作用和奇点结构，仍然远远没有得到很好的理解。在项目的这一部分中，PI的目标是(i)对熵弱解的冲击数量进行定量分析，并研究它们在各种非线性波模型和守恒律的非线性双曲系统中的一般规律性；（ii）详细描述了非局部平衡律的熵弱解的激波形成和破波，并追踪了它们对BV规则性和稳定性结果的影响。我们所追求的方法是利用解的分段正则性，寻求将在低正则性空间上定义的方程简化为在更正则的空间上与有限维流形上的ODE耦合的方程。在已知感兴趣的解是一般分段光滑的情况下，这可能会对数值分析的更广泛领域产生影响，提出新的高阶计算算法。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。