Regularity and Approximation of Solutions to Conservation Laws

守恒定律解的正则性和近似性

• 批准号: 2306926
• 负责人: Alberto Bressan
• 金额: $ 38.78万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306926&HistoricalAwards=false
• 关键词: Regularity; Approximation; Solutions; Conservation; Laws

Hyperbolic conservation laws provide basic mathematical models for continuum physics, and are widely used by scientists and engineers, for instance, in the study of traffic flows and flame propagation fronts. There is a general expectation that these equations should be deterministic: knowing an initial configuration one should be able to uniquely predict the future evolution. However, recent mathematical advances point to the fact that this is not always true. A major goal of this project is to better understand in which situations the uniqueness of solutions can be guaranteed, compared with examples where multiple solutions occur, in one or more space dimensions. Based on these theoretical advances, the investigator will then provide new error bounds for a wide class of computational schemes, which are used in applications as predictive tools. A further research direction will be the accurate description of how solutions can lose regularity. In other words: what happens at the first instant of time when a new shock wave, such as a sudden alteration in pressure, is formed. The project will provide research training opportunities for graduate students and postdoctoral associates. The project will address some fundamental issues at the frontier of the current theory of hyperbolic conservation laws. New uniqueness or non-uniqueness results will be sought, in a wider class of weak solutions, possibly with unbounded variation. For one-dimensional hyperbolic conservation laws endowed with a strictly convex entropy, the investigator aims at establishing universal error estimates, valid for all approximation schemes which are compatible with the conservation equations and the entropy conditions. In addition, for various classes of nonlinear wave equations, a local asymptotic description of generic solutions will be provided, in a neighborhood of a point where a new singularity emerges.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.

双曲线保护定律为连续物理学提供了基本的数学模型，并被科学家和工程师广泛使用，例如在交通流和火焰传播方面的研究中。人们普遍期望这些方程式应该是确定性的：知道初始配置应该能够唯一地预测未来的演变。但是，最近的数学进步指出，这并不总是正确的。该项目的一个主要目标是更好地理解与在一个或多个空间维度中发生多个解决方案的示例相比，可以保证解决方案的独特性。基于这些理论进步，研究者将为广泛的计算方案提供新的错误界限，这些计算方案被用作预测工具。进一步的研究方向将是对解决方案如何失去规律性的准确描述。换句话说：在形成新的冲击波（例如突然改变压力）时，会发生什么。该项目将为研究生和博士后伙伴提供研究培训机会。该项目将解决当前双曲线保护法理论边界的一些基本问题。在更广泛的弱解决方案中，将寻求新的唯一性或非唯一性结果，这些解决方案可能具有无限的变化。对于具有严格凸入熵的一维双曲保护定律，研究者旨在建立通用误差估计，对与保护方程和熵条件兼容的所有近似方案有效。此外，对于各种非线性波方程，将在出现新的奇异性的社区中提供对通用解决方案的本地渐近描述。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来获得支持的。