Stability Theory for Systems of Hyperbolic Conservation Laws

该项目旨在开发数学工具，用于研究双曲守恒定律的稳定性理论，该定律模拟了广泛的物理系统，包括交通流量以及流体和气体动力学。这些系统经常会出现冲击或不连续性，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一致稳定性，特别是Navier-Stokes方程，以及设计方法来减轻流体动力学中粘性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、分析和交流方面的专业知识。该项目将通过扩展带位移的加权收缩理论，进一步发展双曲守恒律理论，以获得弱/BV原理、BV解相对于剧烈初始扰动的稳定性以及物理粘性模型的无粘极限作为Navier-Stokes方程。该项目将建立在加权收缩理论的最新发展的基础上，以解决等熵流情况下的一个20年前的猜想。该项目还将考虑多维设置，其中已知解决方案的唯一性失败。虽然不稳定性是由于湍流引起的，但缺乏唯一性严重质疑了模型本身的预测能力。这种病理学给该领域带来了机遇和严峻的挑战。这项研究将开发一种理论，以协调不稳定性和可预测性的不连续流在高雷诺数。这一奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.