Stability Theory for Systems of Hyperbolic Conservation Laws

该项目旨在开发用于研究双曲守恒定律稳定性理论的数学工具，双曲守恒定律可以模拟广泛的物理系统，包括交通流和流体和气体动力学。这些系统经常产生冲击或不连续，这对它们的数学处理提出了重大挑战。主要目的是研究这些模型的粘性近似的均匀稳定性，特别是Navier-Stokes方程，以及设计方法来减轻流体动力学中粘度对冲击的不稳定影响。该项目还将为本科生、研究生和博士后研究人员提供培训和指导机会，以提高他们在建模、分析和沟通方面的专业知识。本项目将进一步发展双曲守恒定律理论，通过扩展带位移的加权收缩理论，得到弱/BV原理，BV解对野初始扰动的稳定性，以及物理粘性模型的无粘极限，如Navier-Stokes方程。该项目将建立在带位移的加权收缩理论的最新发展上，以解决一个20年前关于等熵流的猜想。该项目还将考虑在解决方案唯一性失败的情况下进行多维设置。虽然湍流会导致不稳定性，但由于缺乏唯一性，模型本身的预测能力受到严重质疑。这种病理给该领域带来了机遇，也带来了严峻的挑战。本研究将发展一种理论来调和高雷诺数下不连续流的不稳定性和可预测性。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 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.