Stability Theory for Systems of Hyperbolic Conservation Laws

双曲守恒定律系统的稳定性理论

• 批准号: 2306852
• 负责人: Alexis Vasseur
• 金额: $ 34万
• 依托单位: University of Texas at Austin
• 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: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306852&HistoricalAwards=false
• 关键词: Stability; Theory; Systems; Hyperbolic; Conservation

This project aims to develop mathematical tools for studying the stability theory of hyperbolic conservation laws, which model a wide range of physical systems, including traffic flow and fluid and gas dynamics. These systems often develop shocks or discontinuities, which pose significant challenges for their mathematical treatment. The primary objective is to investigate the uniform stability of viscous approximations of these models, particularly the Navier-Stokes equation, and design methods to mitigate the destabilizing effect of viscosity on shocks in fluid dynamics. The project will also offer training and mentorship opportunities for undergraduate and graduate students and postdoctoral researchers, to enhance their expertise in modeling, analysis, and communication. This project will further develop the theory for hyperbolic conservation laws, by extending the theory of weighted contraction with shifts, to obtain weak/BV principles, stability of BV solutions with respect to wild initial perturbations, and inviscid limit of physical viscous models as the Navier-Stokes equation. The project will build on recent developments in the theory of weighted contraction with shifts to solve a twenty-year-old conjecture in the case of isentropic flows. The project will also consider multi-D settings where the uniqueness of solutions is known to fail. While instabilities are expected due to turbulence, the lack of uniqueness seriously questions the prediction abilities of the models themselves. This pathology brings both opportunities and formidable challenges to the field. This research will develop a theory to reconcile instability and predictability for discontinuous flows at high Reynolds numbers.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.

该项目旨在开发数学工具来研究双曲线保护定律的稳定性理论，该法律对广泛的物理系统进行了建模，包括交通流量以及流体和气体动力学。这些系统通常会出现冲击或不连续性，这对其数学处理构成了重大挑战。主要目的是研究这些模型粘性近似值的均匀稳定性，尤其是Navier-Stokes方程，以及设计方法，以减轻粘度对流体动力学冲击的不稳定影响。该项目还将为本科生和研究生以及博士后研究人员提供培训和指导机会，以增强他们在建模，分析和沟通方面的专业知识。该项目将通过扩展加权理论，通过转移的加权理论，获得弱/BV原理，BV解决方案在野生初始扰动方面的稳定性以及物理粘性模型的稳定性，作为Navier-Stokes方程。该项目将基于加权理论的最新发展，而在等性流动的情况下，该项目将解决一个二十岁的猜想。该项目还将考虑已知解决方案唯一性失败的多D设置。尽管预计由于动荡而预期不稳定性，但缺乏唯一性会严重质疑模型本身的预测能力。这种病理为该领域带来了机会和巨大的挑战。这项研究将开发一种理论，以调和雷诺数高的不连续流动的不稳定性和可预测性。该奖项反映了NSF的法定使命，并被认为是值得通过基金会的智力优点和更广泛影响的评估评估来支持的。